From 457ca7a8da22013fb2f3bf8db3103938a073c753 Mon Sep 17 00:00:00 2001 From: Kevin Lin Date: Sun, 26 Jul 2026 22:07:48 -0700 Subject: [PATCH 01/16] next init --- stemformulas-next/.gitignore | 41 + stemformulas-next/AGENTS.md | 5 + stemformulas-next/CLAUDE.md | 1 + stemformulas-next/README.md | 36 + stemformulas-next/eslint.config.mjs | 18 + stemformulas-next/next.config.ts | 7 + stemformulas-next/package-lock.json | 6157 +++++++++++++++++++++ stemformulas-next/package.json | 24 + stemformulas-next/src/app/favicon.ico | Bin 0 -> 25931 bytes stemformulas-next/src/app/globals.css | 49 + stemformulas-next/src/app/layout.tsx | 30 + stemformulas-next/src/app/page.module.css | 142 + stemformulas-next/src/app/page.tsx | 66 + stemformulas-next/tsconfig.json | 34 + 14 files changed, 6610 insertions(+) create mode 100644 stemformulas-next/.gitignore create mode 100644 stemformulas-next/AGENTS.md create mode 100644 stemformulas-next/CLAUDE.md create mode 100644 stemformulas-next/README.md create mode 100644 stemformulas-next/eslint.config.mjs create mode 100644 stemformulas-next/next.config.ts create mode 100644 stemformulas-next/package-lock.json create mode 100644 stemformulas-next/package.json create mode 100644 stemformulas-next/src/app/favicon.ico create mode 100644 stemformulas-next/src/app/globals.css create mode 100644 stemformulas-next/src/app/layout.tsx create mode 100644 stemformulas-next/src/app/page.module.css create mode 100644 stemformulas-next/src/app/page.tsx create mode 100644 stemformulas-next/tsconfig.json diff --git a/stemformulas-next/.gitignore b/stemformulas-next/.gitignore new file mode 100644 index 00000000..5ef6a520 --- /dev/null +++ b/stemformulas-next/.gitignore @@ -0,0 +1,41 @@ +# See https://help.github.com/articles/ignoring-files/ for more about ignoring files. + +# dependencies +/node_modules +/.pnp +.pnp.* +.yarn/* +!.yarn/patches +!.yarn/plugins +!.yarn/releases +!.yarn/versions + +# testing +/coverage + +# next.js +/.next/ +/out/ + +# production +/build + +# misc +.DS_Store +*.pem + +# debug +npm-debug.log* +yarn-debug.log* +yarn-error.log* +.pnpm-debug.log* + +# env files (can opt-in for committing if needed) +.env* + +# vercel +.vercel + +# typescript +*.tsbuildinfo +next-env.d.ts diff --git a/stemformulas-next/AGENTS.md b/stemformulas-next/AGENTS.md new file mode 100644 index 00000000..8bd0e390 --- /dev/null +++ b/stemformulas-next/AGENTS.md @@ -0,0 +1,5 @@ + +# This is NOT the Next.js you know + +This version has breaking changes — APIs, conventions, and file structure may all differ from your training data. Read the relevant guide in `node_modules/next/dist/docs/` before writing any code. Heed deprecation notices. + diff --git a/stemformulas-next/CLAUDE.md b/stemformulas-next/CLAUDE.md new file mode 100644 index 00000000..43c994c2 --- /dev/null +++ b/stemformulas-next/CLAUDE.md @@ -0,0 +1 @@ +@AGENTS.md diff --git a/stemformulas-next/README.md b/stemformulas-next/README.md new file mode 100644 index 00000000..e215bc4c --- /dev/null +++ b/stemformulas-next/README.md @@ -0,0 +1,36 @@ +This is a [Next.js](https://nextjs.org) project bootstrapped with [`create-next-app`](https://nextjs.org/docs/app/api-reference/cli/create-next-app). + +## Getting Started + +First, run the development server: + +```bash +npm run dev +# or +yarn dev +# or +pnpm dev +# or +bun dev +``` + +Open [http://localhost:3000](http://localhost:3000) with your browser to see the result. + +You can start editing the page by modifying `app/page.tsx`. The page auto-updates as you edit the file. + +This project uses [`next/font`](https://nextjs.org/docs/app/building-your-application/optimizing/fonts) to automatically optimize and load [Geist](https://vercel.com/font), a new font family for Vercel. + +## Learn More + +To learn more about Next.js, take a look at the following resources: + +- [Next.js Documentation](https://nextjs.org/docs) - learn about Next.js features and API. +- [Learn Next.js](https://nextjs.org/learn) - an interactive Next.js tutorial. + +You can check out [the Next.js GitHub repository](https://github.com/vercel/next.js) - your feedback and contributions are welcome! + +## Deploy on Vercel + +The easiest way to deploy your Next.js app is to use the [Vercel Platform](https://vercel.com/new?utm_medium=default-template&filter=next.js&utm_source=create-next-app&utm_campaign=create-next-app-readme) from the creators of Next.js. + +Check out our [Next.js deployment documentation](https://nextjs.org/docs/app/building-your-application/deploying) for more details. diff --git a/stemformulas-next/eslint.config.mjs b/stemformulas-next/eslint.config.mjs new file mode 100644 index 00000000..05e726d1 --- /dev/null +++ b/stemformulas-next/eslint.config.mjs @@ -0,0 +1,18 @@ +import { defineConfig, globalIgnores } from "eslint/config"; +import nextVitals from "eslint-config-next/core-web-vitals"; +import nextTs from "eslint-config-next/typescript"; + +const eslintConfig = defineConfig([ + ...nextVitals, + ...nextTs, + // Override default ignores of eslint-config-next. + globalIgnores([ + // Default ignores of eslint-config-next: + ".next/**", + "out/**", + "build/**", + "next-env.d.ts", + ]), +]); + +export default eslintConfig; diff --git a/stemformulas-next/next.config.ts b/stemformulas-next/next.config.ts new file mode 100644 index 00000000..e9ffa308 --- /dev/null +++ b/stemformulas-next/next.config.ts @@ -0,0 +1,7 @@ +import type { NextConfig } from "next"; + +const nextConfig: NextConfig = { + /* config options here */ +}; + +export default nextConfig; diff --git a/stemformulas-next/package-lock.json b/stemformulas-next/package-lock.json new file mode 100644 index 00000000..c88b943e --- /dev/null +++ b/stemformulas-next/package-lock.json @@ -0,0 +1,6157 @@ +{ + "name": "stemformulas-next", + "version": "0.1.0", + "lockfileVersion": 3, + "requires": true, + "packages": { + "": { + "name": "stemformulas-next", + "version": "0.1.0", + "dependencies": { + "next": "16.2.12", + "react": "19.2.4", + "react-dom": "19.2.4" + }, + "devDependencies": { + "@types/node": "^20", + "@types/react": "^19", + "@types/react-dom": "^19", + "eslint": "^9", + "eslint-config-next": "16.2.12", + "typescript": "^5" + } + }, + "node_modules/@babel/code-frame": { + "version": "7.29.7", + "resolved": "https://registry.npmjs.org/@babel/code-frame/-/code-frame-7.29.7.tgz", + "integrity": 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+ text-decoration: none; +} + +@media (prefers-color-scheme: dark) { + html { + color-scheme: dark; + } +} diff --git a/stemformulas-next/src/app/layout.tsx b/stemformulas-next/src/app/layout.tsx new file mode 100644 index 00000000..1da6a2fa --- /dev/null +++ b/stemformulas-next/src/app/layout.tsx @@ -0,0 +1,30 @@ +import type { Metadata } from "next"; +import { Geist, Geist_Mono } from "next/font/google"; +import "./globals.css"; + +const geistSans = Geist({ + variable: "--font-geist-sans", + subsets: ["latin"], +}); + +const geistMono = Geist_Mono({ + variable: "--font-geist-mono", + subsets: ["latin"], +}); + +export const metadata: Metadata = { + title: "Create Next App", + description: "Generated by create next app", +}; + +export default function RootLayout({ + children, +}: Readonly<{ + children: React.ReactNode; +}>) { + return ( + + {children} + + ); +} diff --git a/stemformulas-next/src/app/page.module.css b/stemformulas-next/src/app/page.module.css new file mode 100644 index 00000000..643fe27d --- /dev/null +++ b/stemformulas-next/src/app/page.module.css @@ -0,0 +1,142 @@ +.page { + --background: #fafafa; + --foreground: #fff; + + --text-primary: #000; + --text-secondary: #666; + + --button-primary-hover: #383838; + --button-secondary-hover: #f2f2f2; + --button-secondary-border: #ebebeb; + + display: flex; + flex: 1; + flex-direction: column; + align-items: center; + justify-content: center; + font-family: var(--font-geist-sans); + background-color: var(--background); +} + +.main { + display: flex; + flex: 1; + width: 100%; + max-width: 800px; + flex-direction: column; + align-items: flex-start; + justify-content: space-between; + background-color: var(--foreground); + padding: 120px 60px; +} + +.intro { + display: flex; + flex-direction: column; + align-items: flex-start; + text-align: left; + gap: 24px; +} + +.intro h1 { + max-width: 320px; + font-size: 40px; + font-weight: 600; + line-height: 48px; + letter-spacing: -2.4px; + text-wrap: balance; + color: var(--text-primary); +} + +.intro p { + max-width: 440px; + font-size: 18px; + line-height: 32px; + text-wrap: balance; + color: var(--text-secondary); +} + +.intro a { + font-weight: 500; + color: var(--text-primary); +} + +.ctas { + display: flex; + flex-direction: row; + width: 100%; + max-width: 440px; + gap: 16px; + font-size: 14px; +} + +.ctas a { + display: flex; + justify-content: center; + align-items: center; + height: 40px; + padding: 0 16px; + border-radius: 128px; + border: 1px solid transparent; + transition: 0.2s; + cursor: pointer; + width: fit-content; + font-weight: 500; +} + +a.primary { + background: var(--text-primary); + color: var(--background); + gap: 8px; +} + +a.secondary { + border-color: var(--button-secondary-border); +} + +/* Enable hover only on non-touch devices */ +@media (hover: hover) and (pointer: fine) { + a.primary:hover { + background: var(--button-primary-hover); + border-color: transparent; + } + + a.secondary:hover { + background: var(--button-secondary-hover); + border-color: transparent; + } +} + +@media (max-width: 600px) { + .main { + padding: 48px 24px; + } + + .intro { + gap: 16px; + } + + .intro h1 { + font-size: 32px; + line-height: 40px; + letter-spacing: -1.92px; + } +} + +@media (prefers-color-scheme: dark) { + .logo { + filter: invert(); + } + + .page { + --background: #000; + --foreground: #000; + + --text-primary: #ededed; + --text-secondary: #999; + + --button-primary-hover: #ccc; + --button-secondary-hover: #1a1a1a; + --button-secondary-border: #1a1a1a; + } +} diff --git a/stemformulas-next/src/app/page.tsx b/stemformulas-next/src/app/page.tsx new file mode 100644 index 00000000..7b947a2d --- /dev/null +++ b/stemformulas-next/src/app/page.tsx @@ -0,0 +1,66 @@ +import Image from "next/image"; +import styles from "./page.module.css"; + +export default function Home() { + return ( +
+
+ Next.js logo +
+

To get started, edit the page.tsx file.

+

+ Looking for a starting point or more instructions? Head over to{" "} + + Templates + {" "} + or the{" "} + + Learning + {" "} + center. +

+
+ +
+
+ ); +} diff --git a/stemformulas-next/tsconfig.json b/stemformulas-next/tsconfig.json new file mode 100644 index 00000000..cf9c65d3 --- /dev/null +++ b/stemformulas-next/tsconfig.json @@ -0,0 +1,34 @@ +{ + "compilerOptions": { + "target": "ES2017", + "lib": ["dom", "dom.iterable", "esnext"], + "allowJs": true, + "skipLibCheck": true, + "strict": true, + "noEmit": true, + "esModuleInterop": true, + "module": "esnext", + "moduleResolution": "bundler", + "resolveJsonModule": true, + "isolatedModules": true, + "jsx": "react-jsx", + "incremental": true, + "plugins": [ + { + "name": "next" + } + ], + "paths": { + "@/*": ["./src/*"] + } + }, + "include": [ + "next-env.d.ts", + "**/*.ts", + "**/*.tsx", + ".next/types/**/*.ts", + ".next/dev/types/**/*.ts", + "**/*.mts" + ], + "exclude": ["node_modules"] +} From 82106717875549e3318cbb0b104e8169e2cefaaf Mon Sep 17 00:00:00 2001 From: Kevin Lin Date: Sun, 26 Jul 2026 23:32:11 -0700 Subject: [PATCH 02/16] working formula page --- .gitignore | 3 +- 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mode 100644 stemformulas-next/src/content/formulas/sphere-equation.mdx create mode 100644 stemformulas-next/src/content/formulas/spring-frequency-period.mdx create mode 100644 stemformulas-next/src/content/formulas/stokes-theorem.mdx create mode 100644 stemformulas-next/src/content/formulas/taylor-expansion-of-e.mdx create mode 100644 stemformulas-next/src/content/formulas/taylor-maclaurin-series.mdx create mode 100644 stemformulas-next/src/content/formulas/thin-lens-equation.mdx create mode 100644 stemformulas-next/src/content/formulas/triangle.png create mode 100644 stemformulas-next/src/content/formulas/trig-angle-sum-difference-identities.mdx create mode 100644 stemformulas-next/src/content/formulas/trig-in-terms-of-e.mdx create mode 100644 stemformulas-next/src/content/formulas/vector-calculus-product-rules.mdx create mode 100644 stemformulas-next/src/content/formulas/viscosity.mdx diff --git a/.gitignore b/.gitignore index 4d0de7e2..32e88ff1 100644 --- a/.gitignore +++ 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"https://github.com/sponsors/wooorm" + } + } + } +} diff --git a/package.json b/package.json new file mode 100644 index 00000000..a877b703 --- /dev/null +++ b/package.json @@ -0,0 +1,10 @@ +{ + "dependencies": { + "@mdx-js/mdx": "^3.1.1", + "@mdx-js/react": "^3.1.1", + "@next/mdx": "^16.2.12", + "gray-matter": "^4.0.3", + "rehype-katex": "^7.0.1", + "remark-math": "^6.0.0" + } +} diff --git a/stemformulas-next/.gitignore b/stemformulas-next/.gitignore index 5ef6a520..c427d186 100644 --- a/stemformulas-next/.gitignore +++ b/stemformulas-next/.gitignore @@ -1,7 +1,7 @@ # See https://help.github.com/articles/ignoring-files/ for more about ignoring files. # dependencies -/node_modules +**node_modules** /.pnp .pnp.* .yarn/* diff --git a/stemformulas-next/mdx-components.tsx b/stemformulas-next/mdx-components.tsx new file mode 100644 index 00000000..e7013b7d --- /dev/null +++ b/stemformulas-next/mdx-components.tsx @@ -0,0 +1,5 @@ +import type { MDXComponents } from "mdx/types"; + +export function useMDXComponents(components: MDXComponents): MDXComponents { + return components; +} \ No newline at end of file diff --git a/stemformulas-next/next.config.ts b/stemformulas-next/next.config.ts index e9ffa308..292afd74 100644 --- a/stemformulas-next/next.config.ts +++ b/stemformulas-next/next.config.ts @@ -1,7 +1,9 @@ -import type { NextConfig } from "next"; +import type {NextConfig} from "next"; const nextConfig: NextConfig = { - /* config options here */ + turbopack: { + root: process.cwd(), + }, }; -export default nextConfig; +export default nextConfig \ No newline at end of file diff --git a/stemformulas-next/package-lock.json b/stemformulas-next/package-lock.json index c88b943e..5b250ea3 100644 --- a/stemformulas-next/package-lock.json +++ b/stemformulas-next/package-lock.json @@ -9,8 +9,11 @@ "version": "0.1.0", "dependencies": { "next": "16.2.12", + "next-mdx-remote": "^6.0.0", "react": "19.2.4", - "react-dom": "19.2.4" + "react-dom": "19.2.4", + "rehype-katex": 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"^2.1.0" } }, + "node_modules/vfile": { + "version": "6.0.3", + "resolved": "https://registry.npmjs.org/vfile/-/vfile-6.0.3.tgz", + "integrity": "sha512-KzIbH/9tXat2u30jf+smMwFCsno4wHVdNmzFyL+T/L3UGqqk6JKfVqOFOZEpZSHADH1k40ab6NUIXZq422ov3Q==", + "license": "MIT", + "dependencies": { + "@types/unist": "^3.0.0", + "vfile-message": "^4.0.0" + }, + "funding": { + "type": "opencollective", + "url": "https://opencollective.com/unified" + } + }, + "node_modules/vfile-location": { + "version": "5.0.3", + "resolved": "https://registry.npmjs.org/vfile-location/-/vfile-location-5.0.3.tgz", + "integrity": "sha512-5yXvWDEgqeiYiBe1lbxYF7UMAIm/IcopxMHrMQDq3nvKcjPKIhZklUKL+AE7J7uApI4kwe2snsK+eI6UTj9EHg==", + "license": "MIT", + "dependencies": { + "@types/unist": "^3.0.0", + "vfile": "^6.0.0" + }, + "funding": { + "type": "opencollective", + "url": "https://opencollective.com/unified" + } + }, + "node_modules/vfile-matter": { + "version": "5.0.1", + "resolved": "https://registry.npmjs.org/vfile-matter/-/vfile-matter-5.0.1.tgz", + "integrity": "sha512-o6roP82AiX0XfkyTHyRCMXgHfltUNlXSEqCIS80f+mbAyiQBE2fxtDVMtseyytGx75sihiJFo/zR6r/4LTs2Cw==", + "license": "MIT", + "dependencies": { + "vfile": "^6.0.0", + "yaml": "^2.0.0" + }, + "funding": { + "type": "opencollective", + "url": "https://opencollective.com/unified" + } + }, + "node_modules/vfile-message": { + "version": "4.0.3", + "resolved": "https://registry.npmjs.org/vfile-message/-/vfile-message-4.0.3.tgz", + "integrity": "sha512-QTHzsGd1EhbZs4AsQ20JX1rC3cOlt/IWJruk893DfLRr57lcnOeMaWG4K0JrRta4mIJZKth2Au3mM3u03/JWKw==", + "license": "MIT", + "dependencies": { + "@types/unist": "^3.0.0", + "unist-util-stringify-position": "^4.0.0" + }, + "funding": { + "type": "opencollective", + "url": "https://opencollective.com/unified" + } + }, + "node_modules/web-namespaces": { + "version": "2.0.1", + "resolved": "https://registry.npmjs.org/web-namespaces/-/web-namespaces-2.0.1.tgz", + "integrity": "sha512-bKr1DkiNa2krS7qxNtdrtHAmzuYGFQLiQ13TsorsdT6ULTkPLKuu5+GsFpDlg6JFjUTwX2DyhMPG2be8uPrqsQ==", + "license": "MIT", + "funding": { + "type": "github", + "url": "https://github.com/sponsors/wooorm" + } + }, "node_modules/which": { "version": "2.0.2", "resolved": "https://registry.npmjs.org/which/-/which-2.0.2.tgz", @@ -6117,6 +8117,21 @@ "dev": true, "license": "ISC" }, + "node_modules/yaml": { + "version": "2.9.0", + "resolved": "https://registry.npmjs.org/yaml/-/yaml-2.9.0.tgz", + "integrity": "sha512-2AvhNX3mb8zd6Zy7INTtSpl1F15HW6Wnqj0srWlkKLcpYl/gMIMJiyuGq2KeI2YFxUPjdlB+3Lc10seMLtL4cA==", + "license": "ISC", + "bin": { + "yaml": "bin.mjs" + }, + "engines": { + "node": ">= 14.6" + }, + "funding": { + "url": "https://github.com/sponsors/eemeli" + } + }, "node_modules/yocto-queue": { "version": "0.1.0", "resolved": "https://registry.npmjs.org/yocto-queue/-/yocto-queue-0.1.0.tgz", @@ -6152,6 +8167,16 @@ "peerDependencies": { "zod": "^3.25.0 || ^4.0.0" } + }, + "node_modules/zwitch": { + "version": "2.0.4", + "resolved": "https://registry.npmjs.org/zwitch/-/zwitch-2.0.4.tgz", + "integrity": "sha512-bXE4cR/kVZhKZX/RjPEflHaKVhUVl85noU3v6b8apfQEc1x4A+zBxjZ4lN8LqGd6WZ3dl98pY4o717VFmoPp+A==", + "license": "MIT", + "funding": { + "type": "github", + "url": "https://github.com/sponsors/wooorm" + } } } } diff --git a/stemformulas-next/package.json b/stemformulas-next/package.json index fc5777e6..016b3aad 100644 --- a/stemformulas-next/package.json +++ b/stemformulas-next/package.json @@ -3,15 +3,18 @@ "version": "0.1.0", "private": true, "scripts": { - "dev": "next dev", + "dev": "next dev -p 8501", "build": "next build", "start": "next start", "lint": "eslint" }, "dependencies": { "next": "16.2.12", + "next-mdx-remote": "^6.0.0", "react": "19.2.4", - "react-dom": "19.2.4" + "react-dom": "19.2.4", + "rehype-katex": "^7.0.1", + "remark-math": "^6.0.0" }, "devDependencies": { "@types/node": "^20", diff --git a/stemformulas-next/src/app/formulas/[slug]/layout.tsx b/stemformulas-next/src/app/formulas/[slug]/layout.tsx new file mode 100644 index 00000000..a3e5f7c5 --- /dev/null +++ b/stemformulas-next/src/app/formulas/[slug]/layout.tsx @@ -0,0 +1,7 @@ +export default function FormulaLayout({ + children, +}: { + children: React.ReactNode; +}) { + return children; +} \ No newline at end of file diff --git a/stemformulas-next/src/app/formulas/[slug]/page.module.css b/stemformulas-next/src/app/formulas/[slug]/page.module.css new file mode 100644 index 00000000..b30f6203 --- /dev/null +++ b/stemformulas-next/src/app/formulas/[slug]/page.module.css @@ -0,0 +1,84 @@ +.articleHeader { + max-width: 65ch; + margin-bottom: 0; +} + +.articleLayout { + display: flex; + flex-direction: column; + max-width: 100%; + margin-top: 0; +} + +@media (min-width: 1024px) { + .articleLayout { + flex-direction: row; + } +} + +.tocSidebar { + padding: 0; +} + +@media (min-width: 1024px) { + .tocSidebar { + order: 999; + max-width: 20rem; + padding-left: 2rem; + } +} + +.tocSticky { + position: sticky; + top: 2.5rem; + padding-right: 1.25rem; +} + +.tocDetails { + margin-left: -1.25rem; + margin-top: 0; + overflow: hidden; + border-radius: 0.5rem; + padding-left: 1.25rem; +} + +.tocSummary { + display: block; + cursor: pointer; + background: var(--color-neutral-100); + padding: 0.25rem 1.25rem; + font-size: 1.125rem; + font-weight: 600; + color: var(--color-neutral-800); + margin-left: -1.25rem; +} + +@media (min-width: 1024px) { + .tocSummary { + display: none; + } +} + +.tocContent { + border-left: 1px dotted var(--color-neutral-300); + padding: 0.5rem 1.25rem; + margin-left: -1.25rem; +} + +.contentArea { + min-width: 0; + min-height: 0; + max-width: 65ch; + flex-grow: 1; +} + +.tags { + margin: 0.5rem 0; + font-size: 0.75rem; + color: var(--color-neutral-500); +} + +.articleFooter { + padding-top: 2rem; + max-width: 65ch; +} diff --git a/stemformulas-next/src/app/formulas/[slug]/page.tsx b/stemformulas-next/src/app/formulas/[slug]/page.tsx new file mode 100644 index 00000000..35f0f343 --- /dev/null +++ b/stemformulas-next/src/app/formulas/[slug]/page.tsx @@ -0,0 +1,116 @@ +import { notFound } from "next/navigation"; +import fs from "fs"; +import path from "path"; +import { compileMDX } from "next-mdx-remote/rsc"; +import remarkMath from "remark-math"; +import rehypeKatex from "rehype-katex"; +import Link from "next/link"; +import styles from "./page.module.css"; + +async function getFormula(slug: string) { + const filePath = path.join( + process.cwd(), + "src/content/formulas", + `${slug}.mdx` + ); + + if (!fs.existsSync(filePath)) return null; + + const source = fs.readFileSync(filePath, "utf-8"); + const contentDir = path.join(process.cwd(), "src/content/formulas"); + + function resolveImage(src: string | undefined) { + if (!src || src.startsWith("http")) return src; + const imgPath = path.join(contentDir, src); + if (!fs.existsSync(imgPath)) return src; + const buf = fs.readFileSync(imgPath); + const ext = path.extname(src).slice(1) || "png"; + return `data:image/${ext};base64,${buf.toString("base64")}`; + } + + const { content, frontmatter } = await compileMDX<{ + title: string; + description: string; + latex: string; + tags: string[]; + }>({ + source, + options: { + parseFrontmatter: true, + mdxOptions: { + remarkPlugins: [remarkMath], + rehypePlugins: [rehypeKatex], + }, + }, + components: { + img: ({ src, alt, ...rest }: React.ImgHTMLAttributes) => ( + // eslint-disable-next-line @next/next/no-img-element + {alt + ), + }, + }); + + return { content, frontmatter }; +} + +export async function generateStaticParams() { + const formulasDir = path.join(process.cwd(), "src/content/formulas"); + const files = fs.readdirSync(formulasDir); + + return files + .filter((file) => file.endsWith(".mdx")) + .map((file) => ({ + slug: file.replace(".mdx", ""), + })); +} + +export default async function FormulaPage({ + params, +}: { + params: Promise<{ slug: string }>; +}) { + const { slug } = await params; + const formula = await getFormula(slug); + + if (!formula) notFound(); + + return ( +
+
+

{formula.frontmatter.title}

+ {formula.frontmatter.tags && formula.frontmatter.tags.length > 0 && ( +
+ {formula.frontmatter.tags.map((tag) => ( + + {tag} + + ))} +
+ )} +
+ +
+
+
+
+ + Table of Contents + +
+ {/* TOC will be populated later */} +
+
+
+
+ +
+ {formula.content} +
+
+ +
+ {/* Prev/next navigation will go here */} +
+
+ ); +} diff --git a/stemformulas-next/src/app/globals.css b/stemformulas-next/src/app/globals.css index 4c18fb3b..460ea211 100644 --- a/stemformulas-next/src/app/globals.css +++ b/stemformulas-next/src/app/globals.css @@ -1,40 +1,42 @@ :root { - --background: #ffffff; - --foreground: #171717; -} - -@media (prefers-color-scheme: dark) { - :root { - --background: #0a0a0a; - --foreground: #ededed; - } + --color-neutral-50: #f9fafb; + --color-neutral-100: #f3f4f6; + --color-neutral-200: #e5e7eb; + --color-neutral-300: #d1d5db; + --color-neutral-400: #9ca3af; + --color-neutral-500: #6b7280; + --color-neutral-600: #4b5563; + --color-neutral-700: #374151; + --color-neutral-800: #1f2937; + --color-neutral-900: #111827; + --color-primary-500: #64748b; + --color-primary-600: #475569; + --color-primary-700: #334155; } html { + scroll-behavior: smooth; height: 100%; } -html, -body { - max-width: 100vw; - overflow-x: hidden; -} - body { min-height: 100%; display: flex; flex-direction: column; - color: var(--foreground); - background: var(--background); - font-family: Arial, Helvetica, sans-serif; + margin: 0 auto; + padding: 0 1.5rem; + font-family: ui-sans-serif, system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", + Roboto, "Helvetica Neue", Arial, "Noto Sans", sans-serif; + font-size: 1.125rem; + line-height: 1.75rem; + color: var(--color-neutral-900); + background: #fff; + max-width: 80rem; -webkit-font-smoothing: antialiased; - -moz-osx-font-smoothing: grayscale; } * { box-sizing: border-box; - padding: 0; - margin: 0; } a { @@ -42,8 +44,323 @@ a { text-decoration: none; } -@media (prefers-color-scheme: dark) { - html { - color-scheme: dark; +/* ── Header ── */ +.site-header { + padding: 1.5rem 0; + font-weight: 600; +} + +@media (min-width: 640px) { + .site-header { + padding: 2.5rem 0; + } +} + +.site-header nav { + display: flex; + align-items: flex-start; + justify-content: space-between; +} + +@media (min-width: 640px) { + .site-header nav { + align-items: center; + } +} + +.site-header .brand { + display: flex; + flex-direction: row; + align-items: center; +} + +.site-header .brand a { + margin-right: 0.5rem; +} + +.site-header .brand img { + width: 32px; + height: 32px; + max-height: 10rem; + max-width: 10rem; + object-fit: scale-down; + object-position: left; +} + +.site-header .brand .site-title { + text-decoration-color: var(--color-primary-500); +} + +.site-header .brand .site-title:hover { + text-decoration: underline; + text-decoration-thickness: 2px; + text-underline-offset: 2px; +} + +.site-header .nav-links { + list-style: none; + display: flex; + flex-direction: column; + text-align: right; + padding: 0; + margin: 0; +} + +@media (min-width: 640px) { + .site-header .nav-links { + flex-direction: row; + } +} + +.site-header .nav-links li { + margin-bottom: 0.25rem; +} + +@media (min-width: 640px) { + .site-header .nav-links li { + margin-bottom: 0; + margin-right: 1.75rem; + } + .site-header .nav-links li:last-child { + margin-right: 0.125rem; + } +} + +.site-header .nav-links a { + text-decoration-color: var(--color-primary-500); +} + +.site-header .nav-links a:hover { + text-decoration: underline; + text-decoration-thickness: 2px; + text-underline-offset: 2px; +} + +/* ── Search button in header ── */ +.search-toggle { + background: none; + border: none; + cursor: pointer; + font-size: 1rem; + color: inherit; + padding: 0; +} + +.search-toggle:hover { + color: var(--color-primary-600); +} + +/* ── Main content area ── */ +.main-content { + flex-grow: 1; +} + +/* ── Footer ── */ +.site-footer { + padding: 2.5rem 0; +} + +/* ── Prose (article content) ── */ +.prose { + max-width: 65ch; +} + +.prose h1 { + font-size: 2.25rem; + font-weight: 800; + color: var(--color-neutral-900); + margin-top: 0; +} + +.prose h2 { + font-size: 1.5rem; + font-weight: 700; + margin-top: 2rem; + margin-bottom: 0.75rem; +} + +.prose p { + margin-bottom: 1rem; +} + +.prose ul, +.prose ol { + padding-left: 1.5rem; + margin-bottom: 1rem; +} + +.prose ul { + list-style-type: disc; +} + +.prose ol { + list-style-type: decimal; +} + +.prose li { + margin-bottom: 0.25rem; +} + +.prose a { + color: var(--color-primary-600); + text-decoration: underline; +} + +.prose a:hover { + color: var(--color-neutral-900); +} + +.prose img { + max-width: 100%; + height: auto; + border-radius: 0.375rem; + margin: 0 auto; + display: block; +} + +.prose img[alt="diagram"] { + width: 400px; +} + +.prose blockquote { + border-left: 4px solid var(--color-neutral-300); + padding-left: 1rem; + color: var(--color-neutral-500); + margin: 1rem 0; +} + +.prose code { + font-size: 0.875em; + background: var(--color-neutral-100); + padding: 0.125rem 0.375rem; + border-radius: 0.25rem; +} + +.prose pre { + background: var(--color-neutral-800); + color: var(--color-neutral-100); + padding: 1rem; + border-radius: 0.375rem; + overflow-x: auto; + margin-bottom: 1rem; +} + +.prose pre code { + background: none; + padding: 0; + color: inherit; +} + +/* ── Tags ── */ +.tag { + display: inline-block; + font-size: 0.75rem; + line-height: 1rem; + color: var(--color-neutral-500); + border: 1px solid var(--color-neutral-200); + border-radius: 0.375rem; + padding: 0.0625rem 0.5rem; + margin-right: 0.25rem; +} + +.tag:hover { + border-color: #000; + color: var(--color-primary-700); +} + +/* ── Table of Contents ── */ +.toc nav ul { + list-style: none; + padding: 0; +} + +.toc nav li { + margin-bottom: 0.25rem; +} + +.toc nav a { + font-size: 0.875rem; + color: var(--color-neutral-500); +} + +.toc nav a:hover { + color: var(--color-primary-600); +} + +/* ── Prev/Next links ── */ +.article-nav { + padding-top: 2rem; +} + +.article-nav hr { + border: none; + border-top: 1px dotted var(--color-neutral-300); +} + +.article-nav .nav-links-row { + display: flex; + justify-content: space-between; + padding-top: 0.75rem; +} + +.article-nav .nav-link { + display: flex; + align-items: center; + gap: 0.5rem; +} + +.article-nav .nav-link:hover span:last-child { + text-decoration: underline; + text-decoration-color: var(--color-primary-500); +} + +.article-nav .nav-arrow { + color: var(--color-neutral-700); + transition: transform 0.2s; +} + +.article-nav .nav-link:hover .nav-arrow { + color: var(--color-primary-600); +} + +.article-nav .nav-link:hover .nav-arrow-prev { + transform: translateX(-2px); +} + +.article-nav .nav-link:hover .nav-arrow-next { + transform: translateX(2px); +} + +/* ── KaTeX ── */ +.katex-html { + font-size: 1.2em; + color: #000; +} + +.katex-display > .katex { + padding-left: 0.5em !important; + text-align: left; +} + +.katex { + padding-left: 0 !important; +} + +/* formula grid */ +.grid-container { + display: grid; + grid-template-columns: 1fr; + gap: 15px; +} + +@media (min-width: 640px) { + .grid-container { + grid-template-columns: 1fr 1fr; + } +} + +@media (min-width: 1024px) { + .grid-container { + grid-template-columns: 1fr 1fr 1fr; } } diff --git a/stemformulas-next/src/app/layout.tsx b/stemformulas-next/src/app/layout.tsx index 1da6a2fa..5c85419a 100644 --- a/stemformulas-next/src/app/layout.tsx +++ b/stemformulas-next/src/app/layout.tsx @@ -1,20 +1,12 @@ import type { Metadata } from "next"; -import { Geist, Geist_Mono } from "next/font/google"; import "./globals.css"; - -const geistSans = Geist({ - variable: "--font-geist-sans", - subsets: ["latin"], -}); - -const geistMono = Geist_Mono({ - variable: "--font-geist-mono", - subsets: ["latin"], -}); +import "katex/dist/katex.min.css"; +import Link from "next/link"; export const metadata: Metadata = { - title: "Create Next App", - description: "Generated by create next app", + title: "stemformulas", + description: + "Stemformulas is the eventual home of every STEM formula in the world.", }; export default function RootLayout({ @@ -23,8 +15,43 @@ export default function RootLayout({ children: React.ReactNode; }>) { return ( - - {children} + + +
+ +
+ +
{children}
+ +
+
+
+
+
+ ); } diff --git a/stemformulas-next/src/content/formulas/amperes-law.mdx b/stemformulas-next/src/content/formulas/amperes-law.mdx new file mode 100644 index 00000000..42a86a24 --- /dev/null +++ b/stemformulas-next/src/content/formulas/amperes-law.mdx @@ -0,0 +1,34 @@ +--- +title: "Ampere's Law" +description: "Ampere's law is a formula that describes the magnetic field produced by a current-carrying wire." +summary: "Ampere's law is a formula that describes the magnetic field produced by a current-carrying wire." +tags: ["physics", "electromagnetism", "Maxwell's equations"] +date: 2022-12-28 +latex: \oint \vec{B} \cdot d\vec{l} = \mu_0 I +--- + +$$ \oint \vec{B} \cdot d\vec{l} = \mu_0 I $$ + +Where +* The left side is a line integral of the magnetic field $\vec{B}$ over a closed loop $\vec{l}$, and +* $\mu_0$ is the [permeability of free space](https://en.wikipedia.org/wiki/Vacuum_permeability), and +* $I$ is the current enclosed by the loop. + +Ampere's law is a formula that describes the magnetic field produced by a current-carrying wire. It says that the magneitc field created by an electric current is proportional to the magnitude of the current. + +## With displacement current + +Ampere's law has a more general form that includes displacement current. The displacement current is proportional to the rate of change of the electric flux. [This site](http://teacher.pas.rochester.edu/phy122/lecture_notes/Chapter35/chapter35.html) has a good explanation of why this modification is needed. The modified form is: + + +$$ \oint \vec{B} \cdot d\vec{l} = \mu_0 I + \mu_0 \epsilon_0 \frac{d\vec{\phi_E}}{dt} $$ + +Where, in addition to the above definitions, +* $\epsilon_0$ is the [permittivity of free space](https://en.wikipedia.org/wiki/Vacuum_permittivity), and +* $\vec{\phi_E}$ is the electric flux intercepted by the surface spanned by the integration path of the magnetic field. + + +## Sources +- [Wikipedia](https://en.wikipedia.org/wiki/Amp%C3%A8re%27s_circuital_law) +- [Byjus](https://byjus.com/physics/amperes-law/) +- [Rochester Edu](http://teacher.pas.rochester.edu/phy122/lecture_notes/Chapter35/chapter35.html) diff --git a/stemformulas-next/src/content/formulas/arithmetic-gradient.mdx b/stemformulas-next/src/content/formulas/arithmetic-gradient.mdx new file mode 100644 index 00000000..ee30dfec --- /dev/null +++ b/stemformulas-next/src/content/formulas/arithmetic-gradient.mdx @@ -0,0 +1,34 @@ +--- +title: "Arithmetic Gradients" +description: "Formulas for calculating the worth of an arithmetic gradient of cash" +summary: "Formulas for calculating the worth of an arithmetic gradient of cash" +tags: ["math", "finance"] +date: 2023-06-26 +latex: \scriptsize P = A \left[\frac{(1+i)^n-1}{i(1+i)^n}\right] + G \left[\frac{(1+i)^n-in-1}{i^2(1+i)^n}\right] +--- + +## Present Worth of an Arithmetic Gradient + +For an arithmetic gradient of costs or benefits where $\small A$ is received in the first period, $\small A + G$ in the second period, $\small A + 2G$ in the third period, and so on, the present worth $\small P$ is given by: + +$$P = A \left[\frac{(1+i)^n-1}{i(1+i)^n}\right] + G \left[\frac{(1+i)^n-in-1}{i^2(1+i)^n}\right]$$ + +Where + +* $\small P$ is the present worth, +* $\small A$ is the constant amount being added to in the gradient, +* $\small G$ is the gradient or the constant increase per period, +* $\small i$ is the interest rate, and +* $\small n$ is the number of periods. + +Note that this is just the sum of A being converted to P and G being converted to P using their respective factors. +## Uniform Series Worth of an Arithmetic Gradient + +It is also possible to convert the linearly increasing series of payments (G) to a uniform series of payments (A) using the following formula: + +$$ A = G \cdot (A/G, i ,n) = G \left[\frac{1}{i} - \frac{n}{(1+i)^n - 1}\right]$$ + + + +## Sources +- [Eastern Mediterranean University (PDF)](https://staff.emu.edu.tr/gokhanizbirak/Documents/courses/ieng323-mane323/assignments-homeworks/Formulas-Engineering-Economy.pdf) diff --git a/stemformulas-next/src/content/formulas/bayes-theorem.mdx b/stemformulas-next/src/content/formulas/bayes-theorem.mdx new file mode 100644 index 00000000..c66cc754 --- /dev/null +++ b/stemformulas-next/src/content/formulas/bayes-theorem.mdx @@ -0,0 +1,38 @@ +--- +title: Bayes' Theorem +description: "The formula for Bayes' Theorem." +summary: "The formula for Bayes' Theorem." +tags: ["mathematics", "probability", "statistics"] +date: 2023-07-16 +latex: "P(A|B) = \\frac{P(B|A)P(A)}{P(B)}" +--- + +Bayes' Theorem is given by the formula: + +$$ P(A|B) = \frac{P(B|A)P(A)}{P(B)} $$ + +Where + +* $\small P(A|B)$ is the probability of event $A$ given event $B$, +* $\small P(B|A)$ is the probability of event $B$ given event $A$, +* $\small P(A)$ is the probability of event $A$, and +* $\small P(B)$ is the probability of event $B$. + +## Expanded form + +The denominator can also be expanded using the law of total probability to give: + +$$ P(A|B) = \frac{P(B|A)P(A)}{P(B|A)P(A) + P(B|\neg A)P(\neg A)} $$ + +Where $\small \neg A$ is the complement of $\small A$. + +## Intuition + +This video by 3Blue1Brown was the single biggest help in my understanding of Bayes' Theorem, you should definitely watch it if you're struggling to understand it: + +{{< youtube id="HZGCoVF3YvM" >}} + +## Sources + +- [Wikipedia](https://en.wikipedia.org/wiki/Bayes%27_theorem) +- [Math is Fun](https://www.mathsisfun.com/data/bayes-theorem.html) \ No newline at end of file diff --git a/stemformulas-next/src/content/formulas/bernoullis-principle.mdx b/stemformulas-next/src/content/formulas/bernoullis-principle.mdx new file mode 100644 index 00000000..0bd99499 --- /dev/null +++ b/stemformulas-next/src/content/formulas/bernoullis-principle.mdx @@ -0,0 +1,49 @@ +--- +title: Bernoulli's Principle +description: "The relation between various sources of pressure in a fluid." +summary: "The relation between various sources of pressure in a fluid." +tags: ["physics", "fluid mechanics"] +date: 2024-02-07 +latex: P + \frac{1}{2} \rho v^2 + \rho g h = \text{const} +--- + +The formula for Bernoulli's Principle (also known as Bernoulli's Equation) states that for two points along the streamline of an incompressible, inviscid, steady-flow fluid, the sum of the pressure, kinetic energy per unit volume, and potential energy per unit volume remains constant. It has the following forms: + +## Pressure Form + +$$ P_1 + \frac{1}{2} \rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho g h_2 $$ + +Where + +* $ \small \small P_1 $ and $ \small \small P_2 $ are the pressures at points 1 and 2, +* $ \small \small \rho $ is the density of the fluid, +* $ \small \small v_1 $ and $ \small \small v_2 $ are the velocities at points 1 and 2, +* $ \small \small g $ is the acceleration due to gravity, and +* $ \small \small h_1 $ and $ \small \small h_2 $ are the heights at points 1 and 2. + +It can be also be written by stating that the sum of the quantities remain constant: + +$$ P + \frac{1}{2} \rho v^2 + \rho g h = \text{const} $$ + + +## Energy Form + +$$ \frac{P}{\rho} + \frac{1}{2} v^2 + g h = \text{const} $$ + +$$ \frac{P_1}{\rho} + \frac{1}{2} v_1^2 + g h_1 = \frac{P_2}{\rho} + \frac{1}{2} v_2^2 + g h_2 $$ + +## Head Form + +$$ \frac{P}{\gamma} + \frac{v^2}{2g} + h = \text{const} $$ + +$$ \frac{P_1}{\gamma} + \frac{v_1^2}{2g} + h_1 = \frac{P_2}{\gamma} + \frac{v_2^2}{2g} + h_2 $$ + +Where + +* $ \small \small \gamma $ is the [specific weight](/formulas/specific-weight/) of the fluid. + + +## Sources + +- [Wikipedia](https://en.wikipedia.org/wiki/Bernoulli%27s_principle) +- [Khan Academy](https://www.khanacademy.org/science/physics/fluids/fluid-dynamics/a/what-is-bernoullis-equation) diff --git a/stemformulas-next/src/content/formulas/binets-formula.mdx b/stemformulas-next/src/content/formulas/binets-formula.mdx new file mode 100644 index 00000000..aa9551c8 --- /dev/null +++ b/stemformulas-next/src/content/formulas/binets-formula.mdx @@ -0,0 +1,20 @@ +--- +title: "Binet's Formula" +description: "The closed form formula for calculating the n-th Fibonacci number." +summary: "The closed form formula for calculating the n-th Fibonacci number." +tags: ["math"] +date: 2025-01-06 +latex: "F_n = \\frac{\\left(\\frac{1+\\sqrt{5}}{2} \\right)^n - \\left(\\frac{1-\\sqrt{5}}{2} \\right)^n}{\\sqrt{5}}" +--- + +Binet's formula allows you to calculate the n-th Fibonacci number, $ \small F_n $, with a closed form solution. It is as follows: + +$$ F_n = \frac{\left(\frac{1+\sqrt{5}}{2} \right)^n - \left(\frac{1-\sqrt{5}}{2} \right)^n}{\sqrt{5}} $$ + +Where + +* $ \small n $ is the index of the Fibonacci number you want to calculate. + +## Sources + +- [Art of Problem Solving](https://artofproblemsolving.com/wiki/index.php/Binet%27s_Formula) diff --git a/stemformulas-next/src/content/formulas/biot-savart-law.mdx b/stemformulas-next/src/content/formulas/biot-savart-law.mdx new file mode 100644 index 00000000..e7b09b18 --- /dev/null +++ b/stemformulas-next/src/content/formulas/biot-savart-law.mdx @@ -0,0 +1,29 @@ +--- +title: Biot-Savart Law +description: "The Biot-Savart law describes the magnetic field produced by a steady current." +summary: "The Biot-Savart law describes the magnetic field produced by a steady current." +tags: ["physics", "electromagnetism"] +date: 2024-04-19 +latex: "B(r) = \\frac{\\mu_0}{4\\pi} \\int \\frac{I d\\ell \\times r'}{|r'|^3}" +--- + +The Biot-Savart Law describes the magnetic field generated by a constant electric current in magneto-statics. It is as follows: + +$$ B(r) = \frac{\mu_0}{4\pi} \int \frac{I d\ell \times r'}{|r'|^3} $$ + +Where + +- $ \small B(r) $ is the magnetic field at point $ \small r $, +- $ \small \mu_0 $ is the permeability of free space, +- $ \small I $ is the current, +- $ \small d\ell $ is an infinitesimal current element, +- $ \small r' = r - \ell $ is the displacement vector from the current element to the observation point, and +- $ \small |r'| $ is the magnitude of $ \small r' $. + + +In other words, to actually use the law, you choose a point r where you want to know the magnetic field, you mathematically describe the geometry of the current-carrying wire such that you know the magnitude and direction of the vector between the current element and the point r, and then you integrate over the entire wire's elements to get the total magnetic field at point r. + + +## Sources + +- [Wikipedia](https://en.wikipedia.org/wiki/Biot%E2%80%93Savart_law) diff --git a/stemformulas-next/src/content/formulas/boltzmann-entropy.mdx b/stemformulas-next/src/content/formulas/boltzmann-entropy.mdx new file mode 100644 index 00000000..dbe5a64b --- /dev/null +++ b/stemformulas-next/src/content/formulas/boltzmann-entropy.mdx @@ -0,0 +1,23 @@ +--- +title: Boltzmann Entropy +description: "The Boltzmann Entropy formula." +summary: "The Boltzmann Entropy formula." +tags: ["physics", "thermodynamics"] +date: 2023-07-23 +latex: "S=k_B\\ln\\Omega" +--- + +The Boltzmann Entropy formula is: + +$$ S = k_B \ln \Omega $$ + +Where + +* $ \small S $ is the Boltzmann Entropy, +* $ \small k_B $ is Boltzmann's constant, and +* $ \small \Omega $ is the number of microstates. + +## Sources + +- [Wikipedia](https://en.wikipedia.org/wiki/Boltzmann%27s_entropy_formula) +- [Chem LibreTexts](https://chem.libretexts.org/Courses/Western_Washington_University/Biophysical_Chemistry_(Smirnov_and_McCarty)/01%3A_Biochemical_Thermodynamics/1.05%3A_The_Boltzmann_Distribution_and_the_Statistical_Definition_of_Entropy) \ No newline at end of file diff --git a/stemformulas-next/src/content/formulas/canonical-partition-function.mdx b/stemformulas-next/src/content/formulas/canonical-partition-function.mdx new file mode 100644 index 00000000..185d7791 --- /dev/null +++ b/stemformulas-next/src/content/formulas/canonical-partition-function.mdx @@ -0,0 +1,63 @@ +--- +title: Canonical Partition Function +description: "The expression describing a canonical ensemble in statistical mechanics." +summary: "The expression describing a canonical ensemble in statistical mechanics." +tags: ["physics", "statistical mechanics"] +date: 2024-03-14 +latex: Z = \sum_{i} e^{-\beta E_i} +--- + +The canonical partition function, denoted by \$ \small Z\ $, is a fundamental concept in statistical mechanics. It is a useful tool that can tell us the properties of the system, such as the probability that the system is in a particular state, the average energy of the system, and the heat capacity of the system. + +For a classical, discrete canonical ensemble at a temperature \$ \small T\ $, the canonical partition function is given by: + +## Partition function + +$$ Z = \sum_{i} e^{-\beta E_i} $$ + +Where: + +- \$ \small Z\ $ is the canonical partition function, +- \$ \small E_i\ $ is the energy of the \$ \small i\ $-th microstate, +- The sum is taken over all possible microstates of the system, and +- \$ \small \beta = \frac{1}{k_B T}\ $ is the thermodynamic beta with \$ \small k_B\ $ being the Boltzmann constant and \$ \small T\ $ being the temperature. + + +## Probability of a state + +The probability of the system having energy E_i is given by: + +$$ P(E_i) = \frac{e^{-\beta E_i}}{Z} $$ + +Where: +- \$ \small P(E_i)\ $ is the probability of the system being in the \$ \small i\ $-th microstate, and +- \$ \small Z\ $ is the canonical partition function described above. + +## Other quantities + +A few useful properties can be derived from the partition function. For example, the average energy of the system is given by: + +$$ \langle E \rangle =\frac{1}{Z}\frac{\partial Z}{\partial \beta} = -\frac{\partial \ln Z}{\partial \beta} = k_B T^2 \frac{\partial \ln Z}{\partial T}$$ + +Next, the expectation value of the energy squared is given by: + +$$ \langle E^2 \rangle = \frac{1}{Z}\frac{\partial^2 Z}{\partial \beta^2} $$ + +Combining these two, we can get the mean square fluctiation of the energy: + +$$ \langle \Delta E^2 \rangle = \langle E^2 \rangle - \langle E \rangle^2 = - \frac{\partial \langle E \rangle}{\partial \beta} $$ +k +Finally, the heat capacity of the system is given by: + +$$ C_v(T) = \frac{1}{k_B T^2} \langle \Delta E^2 \rangle$$ + + + + +## Sources + +- [Wikipedia](https://en.wikipedia.org/wiki/Partition_function_(statistical_mechanics)) +- My professor's slides + +## Examples + diff --git a/stemformulas-next/src/content/formulas/capacitance.mdx b/stemformulas-next/src/content/formulas/capacitance.mdx new file mode 100644 index 00000000..fad1a31a --- /dev/null +++ b/stemformulas-next/src/content/formulas/capacitance.mdx @@ -0,0 +1,31 @@ +--- +title: Capacitance +description: "The capacitance of a parallel plate capacitor" +summary: "The capacitance of a parallel plate capacitor" +tags: ["electronics", "electrical engineering"] +date: 2023-09-06 +latex: "C = \\frac{\\epsilon A}{d}" +--- + +The capacitance of a parallel-plate capacitor is given by: + +$$ C = \frac{\epsilon A}{d} $$ + +Where + +- $ \small C $ is the capacitance of the capacitor, measured in farads (F), +- $ \small \epsilon $ is the permittivity of the dielectric between the plates, in farads per meter (F/m), +- $ \small A $ is the area of the plates in square meters (m2), and +- $ \small d $ is the distance between the plates in meters. + + +## Sources + +- [Electronics Notes](https://www.electronics-notes.com/articles/basic_concepts/capacitance/capacitor-formulas-equations.php) +- [Wikipedia](https://en.wikipedia.org/wiki/Capacitor) + +## See also + +- [Capacitor Charge](/formulas/capacitor-charge/) +- [Capacitor Current](/formulas/capacitor-current/) +- [Capacitor Energy](/formulas/capacitor-energy/) diff --git a/stemformulas-next/src/content/formulas/capacitor-charge.mdx b/stemformulas-next/src/content/formulas/capacitor-charge.mdx new file mode 100644 index 00000000..f0f66225 --- /dev/null +++ b/stemformulas-next/src/content/formulas/capacitor-charge.mdx @@ -0,0 +1,30 @@ +--- +title: Capacitor Charge +description: "The relationship between the voltage across a capacitor and the charge stored on each conductor." +summary: "The relationship between the voltage across a capacitor and the charge stored on each conductor." +tags: ["electronics", "electrical engineering"] +date: 2023-09-06 +latex: "V = \\frac{Q}{C}" +--- + +The relationship between the voltage across a capacitor and the charge stored on each conductor is given by: + +$$ V = \frac{Q}{C} $$ + +Where + +- $ \small V $ represents the voltage across the capacitor, +- $ \small Q $ represents the charge stored on the capacitor, and +- $ \small C $ is the capacitance of the capacitor, usually measured in farads (F). + + +## Sources + +- [Electronics Notes](https://www.electronics-notes.com/articles/basic_concepts/capacitance/capacitor-formulas-equations.php) +- [Wikipedia](https://en.wikipedia.org/wiki/Capacitor) + +## See also + +- [Capacitance](/formulas/capacitance/) +- [Capacitor Current](/formulas/capacitor-current/) +- [Capacitor Energy](/formulas/capacitor-energy/) \ No newline at end of file diff --git a/stemformulas-next/src/content/formulas/capacitor-current.mdx b/stemformulas-next/src/content/formulas/capacitor-current.mdx new file mode 100644 index 00000000..99b0a0d4 --- /dev/null +++ b/stemformulas-next/src/content/formulas/capacitor-current.mdx @@ -0,0 +1,30 @@ +--- +title: Capacitor Current +description: "The relationship between the current through a capacitor and the rate of change of voltage across the capacitor." +summary: "The relationship between the current through a capacitor and the rate of change of voltage across the capacitor." +tags: ["electronics", "electrical engineering"] +date: 2023-09-06 +latex: "I = C \\frac{dV}{dt}" +--- + +The current through a capacitor is given by: + +$$ I = C \frac{dV}{dt} $$ + +Where + +- $ \small I $ is the current through the capacitor in amperes (A), +- $ \small C $ is the capacitance of the capacitor in farads (F), and +- $ \small \frac{dV}{dt} $ is the rate of change of voltage across the capacitor with respect to time (V/s). + + +## Sources + +- [Electronics Notes](https://www.electronics-notes.com/articles/basic_concepts/capacitance/capacitor-formulas-equations.php) +- [Wikipedia](https://en.wikipedia.org/wiki/Capacitor) + +## See also + +- [Capacitance](/formulas/capacitance/) +- [Capacitor Charge](/formulas/capacitor-charge/) +- [Capacitor Energy](/formulas/capacitor-energy/) \ No newline at end of file diff --git a/stemformulas-next/src/content/formulas/capacitor-energy.mdx b/stemformulas-next/src/content/formulas/capacitor-energy.mdx new file mode 100644 index 00000000..8c22a59f --- /dev/null +++ b/stemformulas-next/src/content/formulas/capacitor-energy.mdx @@ -0,0 +1,29 @@ +--- +title: Capacitor Energy +description: "The energy stored in a capacitor." +summary: "The energy stored in a capacitor." +tags: ["electronics", "electrical engineering"] +date: 2023-09-06 +latex: "E = \\frac{1}{2} C V^2" +--- + +The energy stored in a capacitor is given by: + +$$ E = \frac{1}{2} C V^2 $$ + +Where + +- $ \small E $ represents the energy stored in the capacitor, measured in joules (J), +- $ \small C $ is the capacitance of the capacitor, measured in farads (F), and +- $ \small V $ represents the voltage across the capacitor, in volts (V). + +## Sources + +- [Electronics Notes](https://www.electronics-notes.com/articles/basic_concepts/capacitance/capacitor-formulas-equations.php) +- [Wikipedia](https://en.wikipedia.org/wiki/Capacitor) + +## See also + +- [Capacitance](/formulas/capacitance/) +- [Capacitor Charge](/formulas/capacitor-charge/) +- [Capacitor Current](/formulas/capacitor-current/) \ No newline at end of file diff --git a/stemformulas-next/src/content/formulas/cauchys-integral-theorem.mdx b/stemformulas-next/src/content/formulas/cauchys-integral-theorem.mdx new file mode 100644 index 00000000..63d2f643 --- /dev/null +++ b/stemformulas-next/src/content/formulas/cauchys-integral-theorem.mdx @@ -0,0 +1,19 @@ +--- +title: "Cauchy's Integral Theorem" +description: "A theorem used for line integrals of complex functions." +summary: "A theorem used for line integrals of complex functions." +tags: ["math", "complex analysis", "integration"] +date: 2022-12-24 +latex: \oint_C f(z)dz = 0 +--- + +$$ \oint_C f(z)dz = 0 $$ + +Where +* $f(z)$ is a [holomorphic](https://en.wikipedia.org/wiki/Holomorphic_function) complex function, and +* $C$ is a [simple closed curve](https://en.wikipedia.org/wiki/Simple_closed_curve) in the complex plane. + +## Sources +- [Wikipedia](https://en.wikipedia.org/wiki/Cauchy%27s_integral_theorem) +- [Wolfram MathWorld](https://mathworld.wolfram.com/CauchyIntegralTheorem.html) + diff --git a/stemformulas-next/src/content/formulas/conditional-probability.mdx b/stemformulas-next/src/content/formulas/conditional-probability.mdx new file mode 100644 index 00000000..308fd458 --- /dev/null +++ b/stemformulas-next/src/content/formulas/conditional-probability.mdx @@ -0,0 +1,23 @@ +--- +title: Conditional Probability +description: "The formula for conditional probability." +summary: "The formula for conditional probability." +tags: ["probability", "statistics"] +date: 2023-07-16 +latex: "P(E|F) = \\frac{P(E \\cap F)}{P(F)}" +--- + +The conditional probability formula is: + +$$ P(E|F) = \frac{P(E \cap F)}{P(F)} $$ + +Where + +* $\small P(E|F)$ is the probability of event $E$ given event $F$, +* $\small P(E \cap F)$ (the intersection of the two events) is the probability of both events $E$ and $F$ occurring, and +* $\small P(F)$ is the probability of event $F$. + +## Sources + +- [Wikipedia](https://en.wikipedia.org/wiki/Conditional_probability) +- [Yale](http://www.stat.yale.edu/Courses/1997-98/101/condprob.htm) \ No newline at end of file diff --git a/stemformulas-next/src/content/formulas/convolution.mdx b/stemformulas-next/src/content/formulas/convolution.mdx new file mode 100644 index 00000000..b6478b2e --- /dev/null +++ b/stemformulas-next/src/content/formulas/convolution.mdx @@ -0,0 +1,19 @@ +--- +title: Convolution +description: "Formula for the convolution of two functions." +summary: "Formula for the convolution of two functions." +tags: ["math", "integration", "calculus"] +date: 2023-01-18 +latex: (f*g)(t)=\int_{-\infty}^{\infty} f(\tau)g(t-\tau)d\tau +--- + +The formula for the convolution of two functions $\small f$ and $\small g$ is: + +$$ (f*g)(t)=\int_{-\infty}^{\infty} f(\tau)g(t-\tau)d\tau$$ + +Where +* $\tau$ is a dummy variable that is used in the integral only. + +## Sources +- [Wikipedia](https://en.wikipedia.org/wiki/Convolution) +- [Wolfram Mathworld](https://mathworld.wolfram.com/Convolution.html) diff --git a/stemformulas-next/src/content/formulas/cosine-law.mdx b/stemformulas-next/src/content/formulas/cosine-law.mdx new file mode 100644 index 00000000..9acb9263 --- /dev/null +++ b/stemformulas-next/src/content/formulas/cosine-law.mdx @@ -0,0 +1,22 @@ +--- +title: Cosine Law +description: "The formula relating the sides and angles of every triangle." +summary: "The formula relating the sides and angles of every triangle." +tags: ["math", "trigonometry", "cos"] +date: 2022-12-26 +latex: c^2 = a^2 + b^2 - 2 \cdot a \cdot b \cdot \cos(\theta) +--- + +For any triangle: + +![diagram](cosine-law.png) + +The following is true: + +$$ c^2 = a^2 + b^2 - 2 \cdot a \cdot b \cdot \cos(\theta) $$ + +Note that the naming of the sides and angle is arbitrary. The formula is true as long as $\small \theta$ is the angle opposite side c. + +## Sources +- [Wikipedia](https://en.wikipedia.org/wiki/Law_of_cosines) +- [Byju's](https://byjus.com/maths/law-of-cosines/) diff --git a/stemformulas-next/src/content/formulas/cosine-law.png b/stemformulas-next/src/content/formulas/cosine-law.png new file mode 100644 index 0000000000000000000000000000000000000000..b15a14365dbd43d7c56f31977d305f42ba6857a1 GIT binary patch literal 21282 zcmeIacT^SIvo+de5K$Br1PKO$qJV*%5kWGFk_A){$vLMjVn9(O2uKtps3Zj>BO43^ zB}>i-0+KTlzginS=iEEKf8Rgvy*tJoj`Oqa-Mv<=s#$Z+YJYWAd72%JJ1`8RQMf33 z8N*08FpOBAk{te*oqygPhChfLF3Zbc8MTK-;Xg=CrB$Rc?0q2h+I2GcZz|i1Iu01d z`~dwwQQ**w35H$HQ;?Ou;%YF_z57bolH2CYV;|kJKGuZ8LC>B^9S+*@f+uS!VAO7Sd?y!SYtzgX5HJA>!v=@+0HdzRTsubUhqVWa2(j@ z=jSIQbLoouyJ}XS5}wJZh|iL2ySwVv#im+?V~1BH42xbQ6UPm281!-vtYlKb17faR zjc({ku{v61!hb(CB8ETl4y1JGKVy$Zpg*P$m65_9xM50oe+(0o=OcnY`h1;E35V)H@EsR@5kGj=C#qniyduk zx8;kl?SG@};E=s<|Ng2!bHnUMj+DDA8MyY7`3D?Y-ZL~Txmi#`OzDMosgKI*NJ4yK 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Note that this expression can be used for $\small r > 1$ unlike the infinite series formula below. + +If $\small r = 1$, then $\small S_n = an$. + +## Infinite Geometric Series +If $\small r$ satisfies $\small 0 < r < 1$, the sum of the infinite series is + +$$S_{\infty} = \sum_{i=0}^{\infty} ar^i = \frac{a}{1-r}$$ + + +## Sources +- [Wikipedia](https://en.wikipedia.org/wiki/Geometric_series) +- [Amsi.org.au](https://amsi.org.au/ESA_Senior_Years/SeniorTopic1/1d/1d_2content_5.html) \ No newline at end of file diff --git a/stemformulas-next/src/content/formulas/fourier-transform.mdx b/stemformulas-next/src/content/formulas/fourier-transform.mdx new file mode 100644 index 00000000..b19bb2ec --- /dev/null +++ b/stemformulas-next/src/content/formulas/fourier-transform.mdx @@ -0,0 +1,39 @@ +--- +title: Fourier + Inverse Fourier Transform +description: "The Fourier and inverse Fourier transform formulas." +summary: "The Fourier and inverse Fourier transform formulas." +tags: ["math", "signal processing", "optics"] +date: 2023-07-03 +latex: X(\omega) = \int_{-\infty}^{\infty}x(t)e^{-j\omega t}\,dt +--- + +## Fourier Transform +The Fourier transform is used to transform a signal from the time domain to the frequency domain. + +The Fourier transform formula is: + +$$ X(\omega) = \int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt $$ + +Where + +* $ \small X(\omega) $ is the Fourier transform of the function $ \small x(t) $, +* $ \small x(t) $ is the input signal in the time domain, +* $ \small j $ is the imaginary unit, +* $ \small e $ is Euler's number, +* $ \small \omega $ is the angular frequency. + + +## Inverse Fourier Transform +The inverse Fourier transform is used to transform a signal from the frequency domain to the time domain. + +The inverse Fourier transform is: + +$$ x(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty}X(\omega)e^{j\omega t}d\omega $$ + + +Note that there are many conventions for these two transforms. $\small i $ is often used as the imaginary unit, and in spatial contexts, $\small x $ and $\small k $ are often used instead of $\small t $ and $\small \omega $, respectively. + +## Sources + +- [Swarthmore College - LPSA](https://lpsa.swarthmore.edu/Fourier/Xforms/FXformIntro.html) +- [BYJU'S](https://byjus.com/maths/fourier-transform/) \ No newline at end of file diff --git a/stemformulas-next/src/content/formulas/fundmantal-theorem-of-calculus.mdx b/stemformulas-next/src/content/formulas/fundmantal-theorem-of-calculus.mdx new file mode 100644 index 00000000..6f3aa55d --- /dev/null +++ b/stemformulas-next/src/content/formulas/fundmantal-theorem-of-calculus.mdx @@ -0,0 +1,32 @@ +--- +title: Fundamental Theorem of Calculus +description: Fundamental Theorem of Calculus +summary: Fundamental Theorem of Calculus +tags: ["math", "calculus"] +date: 2022-12-23 +latex: \int_a^b f(x) dx = F(b) - F(a) +--- +$$\int_a^b f(x) dx = F(b) - F(a)$$ + +Where +* $f$ is a continuous function on $[a, b]$, +* $F$ is an antiderivative of $f$, +* $a$ is the lower bound of the integral, and +* $b$ is the upper bound of the integral. + +This theorem, also known as the second fundamental theorem of calculus says that the definite integral of a continuous function on a closed interval is equal to the difference between the value of the antiderivative at the upper bound and the value of the antiderivative at the lower bound. + +See Wolfram MathWorld below for the other two fundamental theorems of calculus. + +## Sources +- [Wikipedia](https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus) +- [Wolfram MathWorld](https://mathworld.wolfram.com/FundamentalTheoremsofCalculus.html) + +## Example +Due to his terrible posture, Kevin's back is approximately the shape of x^3 on the interval [0, 3]. Given that his chair is the x axis, what is the area between his back and his chair? + +We know that the area between a curve and the x axis is equal to the definite integral of the curve. We also know that the definite integral of a function is equal to the difference between the value of the antiderivative at the upper bound and the value of the antiderivative at the lower bound. So we can plug in the values to get the answer, knowing that the antiderivative of x^3 is x^4/4: + +$$\int_0^3 x^3 dx = F(3) - F(0) = \frac{(3)^4}{4} - 0 = 20.25$$ + + diff --git a/stemformulas-next/src/content/formulas/future-to-net-present-worth.mdx b/stemformulas-next/src/content/formulas/future-to-net-present-worth.mdx new file mode 100644 index 00000000..599be06b --- /dev/null +++ b/stemformulas-next/src/content/formulas/future-to-net-present-worth.mdx @@ -0,0 +1,22 @@ +--- +title: "Present and Future Value" +description: "Formulas for converting between future worth and net present worth" +summary: "Formulas for converting between future worth and net present worth" +tags: ["math", "finance"] +date: 2023-06-24 +latex: F = P(1 + i)^n +--- + +For a sum of money $P$ invested at a compounding interest rate $i$ for $n$ periods, the future value $F$ is given by: + +$$F = P(1 + i)^n$$ + +Where + +* $P$ is the present value, +* $i$ is the interest rate, and +* $n$ is the number of periods + +## Sources +- [Investopedia](https://www.investopedia.com/terms/f/futurevalue.asp) +- [Wikipedia](https://en.wikipedia.org/wiki/Future_value#Compound_interest) \ No newline at end of file diff --git a/stemformulas-next/src/content/formulas/gauss-law.mdx b/stemformulas-next/src/content/formulas/gauss-law.mdx new file mode 100644 index 00000000..74f68887 --- /dev/null +++ b/stemformulas-next/src/content/formulas/gauss-law.mdx @@ -0,0 +1,31 @@ +--- +title: "Gauss's Law" +description: "Gauss's law is a formula that describes the electric field produced by a charge distribution." +summary: "Gauss's law is a formula that describes the electric field produced by a charge distribution." +tags: ["physics", "electromagnetism", "Maxwell's equations"] +date: 2022-12-28 +latex: \oiint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0} +--- + +$$ \oiint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0} $$ + +Where + +* The left side is a surface integral of the electric field $\vec{E}$ over a closed surface $\vec{A}$, and +* $Q_{enc}$ is the charge enclosed by the surface, and +* $\epsilon_0$ is the [permittivity of free space](https://en.wikipedia.org/wiki/Vacuum_permittivity). + +## Differential form + +Gauss's law can also be written in differential form: + +$$ \nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0} $$ + +Where + +* The left side is the [divergence](https://en.wikipedia.org/wiki/Divergence) of the electric field $\vec{E}$, and +* $\rho$ is the [charge density](https://en.wikipedia.org/wiki/Charge_density). + +## Sources +- [Wikipedia](https://en.wikipedia.org/wiki/Gauss%27s_law) +- [Oregon State](https://books.physics.oregonstate.edu/GSF/maxwell1.html) \ No newline at end of file diff --git a/stemformulas-next/src/content/formulas/gaussian.mdx b/stemformulas-next/src/content/formulas/gaussian.mdx new file mode 100644 index 00000000..25ed9493 --- /dev/null +++ b/stemformulas-next/src/content/formulas/gaussian.mdx @@ -0,0 +1,23 @@ +--- +title: Gaussian/Normal Distribution +description: "The formula for the normal distribution." +summary: "The formula for the normal distribution." +tags: ["math", "statistics", "probability theory"] +date: 2023-07-13 +latex: f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2} +--- + +The formula for the normal distribution as a function of the variable x is: + +$$ \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2} $$ + +Where + +* $ \small e\ $ is Euler's number, +* $ \small \mu\ $ is the mean of the distribution, and +* \$ \small \sigma\ $ is the standard deviation of the distribution. + +## Sources + +- [Wikipedia](https://en.wikipedia.org/wiki/Normal_distribution) +- [Britannica](https://www.britannica.com/topic/uniform-distribution-statistics) \ No newline at end of file diff --git a/stemformulas-next/src/content/formulas/gradient.mdx b/stemformulas-next/src/content/formulas/gradient.mdx new file mode 100644 index 00000000..abb1266b --- /dev/null +++ b/stemformulas-next/src/content/formulas/gradient.mdx @@ -0,0 +1,52 @@ +--- +title: Gradient +description: "The concept of gradient in vector calculus." +summary: "The concept of gradient in vector calculus." +tags: ["math", "vector calculus"] +date: 2023-07-08 +latex: \nabla f = \frac{\partial f}{\partial x} \mathbf{i} + \frac{\partial f}{\partial y} \mathbf{j} + \frac{\partial f}{\partial z} \mathbf{k} +--- + +The gradient represents the direction and magnitude of the steepest ascent of the function \$ \small f $ at any given point. + +## Cartesian Coordinates + +The gradient of a scalar function \$ \small f $ in cartesian coordinates is given by: + +$$ \nabla f = \frac{\partial f}{\partial x} \mathbf{i} + \frac{\partial f}{\partial y} \mathbf{j} + \frac{\partial f}{\partial z} \mathbf{k} $$ + + +Where: +- \$ \small \\nabla $ represents the del operator (also known as the gradient operator), +- \$ \small \mathbf{i}, \mathbf{j}, \mathbf{k} $ are the unit vectors in the x, y, and z directions respectively, and +- $ \partial $ represents the partial derivative operator. + +## Cylindrical Coordinates + +The gradient of a scalar function $ \small f $ in cylindrical coordinates is given by: + +$$ \nabla f = \frac{\partial f}{\partial \rho} \hat{\rho} + \frac{1}{\rho} \frac{\partial f}{\partial \varphi} \hat{\varphi} + \frac{\partial f}{\partial z} \hat{z}$$ + +Where: +- $ \small \rho $ is the radial coordinate, $ \small \varphi $ is the azimuthal coordinate, and $ \small z $ is the vertical coordinate, +- $ \hat{\rho}, \hat{\varphi}, \hat{z} $ are the unit vectors in the $ \small \rho, \varphi, z $ directions respectively. + + +## Spherical Coordinates + +The gradient of a scalar function $ \small f $ in spherical coordinates is given by: + +$$ \nabla f = \frac{\partial f}{\partial r} \hat{r} + \frac{1}{r} \frac{\partial f}{\partial \theta} \hat{\theta} + \frac{1}{r \sin \theta} \frac{\partial f}{\partial \varphi} \hat{\varphi} $$ + +Where: +- $ \small r $ is the radial coordinate, $ \small \theta $ is the polar coordinate, and $ \small \varphi $ is the azimuthal coordinate, and +- $ \hat{r}, \hat{\theta}, \hat{\varphi} $ are the unit vectors in the $ \small r, \theta, \varphi $ directions respectively. + +## Sources + +- [Wikipedia](https://en.wikipedia.org/wiki/Gradient) +- [Profound Physics](https://profoundphysics.com/gradient-in-different-coordinates/) + +## See also +- [Divergence](/formulas/divergence/) +- [Curl](/formulas/curl/) \ No newline at end of file diff --git a/stemformulas-next/src/content/formulas/grand-canonical-partition-function.mdx b/stemformulas-next/src/content/formulas/grand-canonical-partition-function.mdx new file mode 100644 index 00000000..26311f39 --- /dev/null +++ b/stemformulas-next/src/content/formulas/grand-canonical-partition-function.mdx @@ -0,0 +1,24 @@ +--- +title: Grand Canonical Partition Function +description: "The expression describing a grand canonical ensemble in statistical mechanics." +summary: "The expression describing a grand canonical ensemble in statistical mechanics." +tags: ["physics", "statistical mechanics"] +date: 2024-03-14 +latex: \Xi = \sum_{i} e^{\beta ( \mu N_i - E_i)} +--- + +The grand canonical partition function, denoted by $ \small \Xi\ $, is a fundamental concept in statistical mechanics, particularly in the grand canonical ensemble. It represents the sum of the canonical partition functions weighted by the Boltzmann factors over all possible microstates with different numbers of particles. + +$$ \Xi = \sum_{i} e^{\beta ( \mu N_i - E_i)} $$ + +Where: + +- $ \small \Xi $ is the grand canonical partition function, +- $ \small \beta = \frac{1}{k_B T} $ is the thermodynamic beta with $ \small k_B$ being the Boltzmann constant and $ \small T $ being the absolute temperature, +- $ \small \mu $ is the chemical potential of the system, +- $ \small N_i $ is the number of particles in the $ \small i $-th microstate, and +- $ \small E_i $ is the energy of the $ \small i $-th microstate. + +## Sources + +- [Wikipedia](https://en.wikipedia.org/wiki/Partition_function_(statistical_mechanics)#Grand_canonical_partition_function) diff --git a/stemformulas-next/src/content/formulas/gravitational-potential-energy.mdx b/stemformulas-next/src/content/formulas/gravitational-potential-energy.mdx new file mode 100644 index 00000000..09923a41 --- /dev/null +++ b/stemformulas-next/src/content/formulas/gravitational-potential-energy.mdx @@ -0,0 +1,38 @@ +--- +title: "Gravitational Potential Energy" +description: "The potential energy of an object in a gravitation field." +summary: "The potential energy of an object in a gravitation field." +tags: ["physics"] +date: 2022-12-24 +latex: "U = mgh" +--- + +$$ U = mgh $$ + +Where +* $U$ is the gravitational potential energy of the object, +* $m$ is the mass of the object, +* $g$ is the acceleration due to gravity, and +* $h$ is the height of the object above the ground. + +This is an approximation that can be made when the object is small compared to the distance between the object and the ground, and g is fairly constant in the domain of observation. + +## Sources +- [Wikipedia](https://en.wikipedia.org/wiki/Potential_energy#Local_approximation) +- [HyperPhysics](http://hyperphysics.phy-astr.gsu.edu/hbase/gpot.html) + +## Example +Kevin throws a 400,000 kg spaceship straight up from the surface of the earth, with an initial velocity of 100 m/s. After some time, the spaceship reaches its peak height. Assuming no losses due to friction, what is this peak height? + +Since no energy is lost due to air friction, we know that all of the kinetic energy gets converted to potential energy. + +The initial kinetic energy is $\scriptsize E_k = \frac{1}{2}mv^2 = 200,000 * 100^2 = 5.0 \cdot 10^{10} J$ + +The final potential energy is $\scriptsize 5.0 \cdot 10^{10} = mgh = 400,000 * 9.8 * h$ + +So, we can solve for the height of the spaceship: + +$\scriptsize h = \frac{5.0 \cdot 10^{10}}{400,000 * 9.8} = 1.27 \cdot 10^4 m$ + +## See also +- [Kinetic Energy](/formulas/kinetic-energy/) diff --git a/stemformulas-next/src/content/formulas/greek-letters.mdx b/stemformulas-next/src/content/formulas/greek-letters.mdx new file mode 100644 index 00000000..45be4fd9 --- /dev/null +++ b/stemformulas-next/src/content/formulas/greek-letters.mdx @@ -0,0 +1,38 @@ +--- +title: "Greek Letters" +description: "A table of Greek letters and their common uses." +summary: "A table of Greek letters and their common uses." +tags: ["misc"] +date: 2023-09-22 +latex: \Delta \Omega \lambda +showTableOfContents: false +--- + +While not technically formulas, Greek letters are used in many formulas and this reference can be useful for identifying their names/their LaTeX commands. + +| Name | Uppercase | Lowercase | Commonly used for... | +|-----------|-------------|-------------|----------------------------------------------| +| Alpha | $ \Alpha $ | $ \alpha $ | linear expansion, nuclear physics | +| Beta | $ \Beta $ | $ \beta $ | nuclear physics | +| Gamma | $ \Gamma $ | $ \gamma $ | special relativity, gamma function | +| Delta | $ \Delta $ | $ \delta $ | differentials, Dirac delta function | +| Epsilon | $ \Epsilon $ | $ \epsilon/\varepsilon $ | permittivity, small quantities | +| Zeta | $ \Zeta $ | $ \zeta $ | Riemann zeta function | +| Eta | $ \Eta $ | $ \eta $ | efficiency, viscosity | +| Theta | $ \Theta $ | $ \theta $ | angles, time complexity | +| Iota | $ \Iota $ | $ \iota $ | | +| Kappa | $ \Kappa $ | $ \kappa $ | thermal conductivity | +| Lambda | $ \Lambda $ | $ \lambda $ | wavelength, decay constant | +| Mu | $ \Mu $ | $ \mu $ | coefficient of friction, permeability | +| Nu | $ \Nu $ | $ \nu $ | frequency | +| Xi | $ \Xi $ | $ \xi $ | | +| Omicron | $ \Omicron $ | $ \omicron $ | | +| Pi | $ \Pi $ | $ \pi $ | product, mathematical constant | +| Rho | $ \Rho $ | $ \rho $ | resistivity, density | +| Sigma | $ \Sigma $ | $ \sigma $ | sum, standard deviation, conductivity, stress| +| Tau | $ \Tau $ | $ \tau $ | torque, time constant | +| Upsilon | $ \Upsilon $ | $ \upsilon $ | | +| Phi | $ \Phi $ | $ \phi/\varphi $ | magnetic flux, golden ratio, angles | +| Chi | $ \Chi $ | $ \chi $ | electric susceptibility, Chi-squared | +| Psi | $ \Psi $ | $ \psi $ | wave function, angles | +| Omega | $ \Omega $ | $ \omega $ | angular velocity, solid angle, resistance | diff --git a/stemformulas-next/src/content/formulas/greens-theorem.mdx b/stemformulas-next/src/content/formulas/greens-theorem.mdx new file mode 100644 index 00000000..a67b39bc --- /dev/null +++ b/stemformulas-next/src/content/formulas/greens-theorem.mdx @@ -0,0 +1,24 @@ +--- +title: "Green's Theorem" +description: "Green's theorem is a formula that describes the relationship between certain closed path line integrals and line integrals." +summary: "Green's theorem is a formula that describes the relationship between certain closed path line integrals and line integrals." +tags: ["vector calculus", "integration", "multivariable calculus", "calculus"] +date: 2022-12-30 +latex: \small \oint_C Pdx + Qdy = \iint_D \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} dA +--- + +If $\small P$ and $\small Q$ are continuous functions of $\small x$ and $\small y$ over the region $\small D$ with a closed path $\small C$ that bounds $\small D$, then Green's theorem states that + +$$ \oint_C Pdx + Qdy = \iint_D \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} dA $$ + +Honestly, just read [Paul's Online Notes](https://tutorial.math.lamar.edu/classes/calcIII/GreensTheorem.aspx), they're great. + +Green's theorem can be thought of as a 2D version of [Stokes' theorem](/formulas/stokes-theorem). + +## Sources +- [Wikipedia](https://en.wikipedia.org/wiki/Green%27s_theorem) +- [Paul's Online Notes](https://tutorial.math.lamar.edu/classes/calcIII/GreensTheorem.aspx) + +## See also +- [Divergence Theorem](/formulas/divergence-theorem) +- [Stokes' Theorem](/formulas/stokes-theorem) diff --git a/stemformulas-next/src/content/formulas/hookes-law-springs.mdx b/stemformulas-next/src/content/formulas/hookes-law-springs.mdx new file mode 100644 index 00000000..839549fc --- /dev/null +++ b/stemformulas-next/src/content/formulas/hookes-law-springs.mdx @@ -0,0 +1,23 @@ +--- +title: Hooke's Law (Spring Constant) +description: Hooke's Law and the Spring Constant relate the force and displacement of a spring. +summary: Hooke's Law and the Spring Constant relate the force and displacement of a spring. +tags: ["physics", "mechanics"] +date: 2022-12-30 +latex: F = -kx +--- + +$$ F = -kx $$ + +Where +* $F$ is the force applied to stretch or compress the spring, +* $k$ is the spring constant of the spring, and +* $x$ is the displacement of the spring. + +The negative sign is used when the force being considered is the "restoring force". +If the force is the "applied force", then the negative sign is omitted. + +## Sources +- [Wikipedia](https://en.wikipedia.org/wiki/Hooke's_law#Tensional_stress_of_a_uniform_bar) +- [Byju's](https://byjus.com/physics/stress-and-strain/) + diff --git a/stemformulas-next/src/content/formulas/hookes-law-stress-strain.mdx b/stemformulas-next/src/content/formulas/hookes-law-stress-strain.mdx new file mode 100644 index 00000000..d243f918 --- /dev/null +++ b/stemformulas-next/src/content/formulas/hookes-law-stress-strain.mdx @@ -0,0 +1,24 @@ +--- +title: Hooke's Law (Modulus of Elasticity) +description: Hooke's Law and Modulus of Elasticity relate the stress and strain of a material. +summary: Hooke's Law and Modulus of Elasticity relate the stress and strain of a material. +tags: ["physics", "mechanical engineering", "materials engineering"] +date: 2022-12-30 +latex: \frac{F}{A} = E\frac{\Delta L}{L} +--- + +$$ \frac{F}{A} = E\frac{\Delta L}{L} $$ + +Where +* $F$ is the force applied to the material, +* $A$ is the area of the material where force is applied, +* $E$ is the modulus of elasticity (a.ka. Young's modulus) of the material, +* $\Delta L$ is the change in length of the material, and +* $L$ is the original length of the material. + +This version of Hooke's Law is generalized from the [original version](/formulas/hookes-law-springs) used to describe the force of springs. + +## Sources +- [Wikipedia](https://en.wikipedia.org/wiki/Hooke's_law#Tensional_stress_of_a_uniform_bar) +- [Byju's](https://byjus.com/physics/stress-and-strain/) + diff --git a/stemformulas-next/src/content/formulas/hydraulic-diameter.mdx b/stemformulas-next/src/content/formulas/hydraulic-diameter.mdx new file mode 100644 index 00000000..21a29c5e --- /dev/null +++ b/stemformulas-next/src/content/formulas/hydraulic-diameter.mdx @@ -0,0 +1,23 @@ +--- +title: Hydraulic Diameter +description: "The generalized characteristic length scale for flow in pipes." +summary: "The generalized characteristic length scale for flow in pipes." +tags: ["physics", "fluid dynamics"] +date: 2024-03-14 +latex: D_h = \frac{4A_c}{P} +--- + +The hydraulic diameter is a characteristic length scale for fluid flow in non-circular pipes. It replaces the diameter of a circular pipe with an equivalent diameter based on the cross-sectional area and wetted perimeter of the non-circular pipe. It is defined as: + +$$ D_h = \frac{4A_c}{P} $$ + +Where: + +- \$ \small D_h\ $ is the hydraulic diameter, +- \$ \small A_c\ $ is the cross-sectional area of the flow, and +- \$ \small P\ $ is the wetted perimeter of the cross-section. + +## Sources + +- [Wikipedia](https://en.wikipedia.org/wiki/Hydraulic_diameter) +- [Engineering Toolbox](https://www.engineeringtoolbox.com/hydraulic-equivalent-diameter-d_458.html) diff --git a/stemformulas-next/src/content/formulas/hydrostatic-equation.mdx b/stemformulas-next/src/content/formulas/hydrostatic-equation.mdx new file mode 100644 index 00000000..e8c18d5e --- /dev/null +++ b/stemformulas-next/src/content/formulas/hydrostatic-equation.mdx @@ -0,0 +1,30 @@ +--- +title: Hydrostatic Equation +description: "The formula for hydrostatic pressure force." +summary: "The formula for hydrostatic pressure force." +tags: ["physics", "fluid mechanics"] +date: 2024-02-07 +latex: \frac{dP}{dz} = - \rho g +--- + +The hydrostatic pressure in a system is the pressure exerted by a fluid at equilibrium due to the force of gravity. The following equations are used to describe the hydrostatic pressure force: + +$$ \frac{dP}{dz} = - \rho g $$ + +$$ P = P_0 - \rho g z $$ + +$$ \Delta P = \rho g \Delta z $$ + +Where + +* $ \small P $ is the pressure, +* $ \small P_0 $ is the pressure at some reference point, +* $ \small z $ is the height above the reference point, +* $ \small \small \rho $ is the density of the fluid, and +* $ \small \small g $ is the acceleration due to gravity. + +Note that the negative sign means that the pressure decreases with increasing height. + +## Sources + +- [Wikipedia](https://en.wikipedia.org/wiki/Hydrostatic_equilibrium) diff --git a/stemformulas-next/src/content/formulas/hyperbolic-trig-functions.mdx b/stemformulas-next/src/content/formulas/hyperbolic-trig-functions.mdx new file mode 100644 index 00000000..c60638dd --- /dev/null +++ b/stemformulas-next/src/content/formulas/hyperbolic-trig-functions.mdx @@ -0,0 +1,26 @@ +--- +title: Hyperbolic Functions +description: Definitions of the hyperbolic functions +summary: Definitions of the hyperbolic functions +tags: ["math", "trigonometry", "e", "imaginary numbers", "complex analysis"] +date: 2022-12-30 +latex: \sinh(x) = \frac{e^x - e^{-x}}{2} +--- + +$$\small \sinh(x) = \frac{e^x - e^{-x}}{2}, \cosh(x) = \frac{e^x + e^{-x}}{2}, \tanh(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}}$$ + +Where +* $e$ is [Euler's number](https://en.wikipedia.org/wiki/E_(mathematical_constant)) ~= 2.71828, and +* $i$ is the imaginary unit, $\sqrt{-1}$ + +## Complex trig definitions + +The hyperbolic functions can be defined in terms of the trig functions with complex arguments: +* $\sinh(x) = -i \sin(ix)$ +* $\cosh(x) = \cos(ix)$ +* $\tanh(x) = -i \tan(ix)$ + + +## Sources +- [Wikipedia](https://en.wikipedia.org/wiki/Hyperbolic_functions#Definition) +- [Byju's](https://byjus.com/maths/hyperbolic-function/) diff --git a/stemformulas-next/src/content/formulas/ideal-gas-law.mdx b/stemformulas-next/src/content/formulas/ideal-gas-law.mdx new file mode 100644 index 00000000..dc800f2d --- /dev/null +++ b/stemformulas-next/src/content/formulas/ideal-gas-law.mdx @@ -0,0 +1,37 @@ +--- +title: "Ideal Gas Law" +description: "The equation of state of a hypothetical ideal gas." +summary: "The equation of state of a hypothetical ideal gas." +tags: ["chemistry"] +date: 2022-11-27 +latex: PV = nRT +--- +The ideal gas law is given by: + +$$PV = nRT$$ + +Where +- $P$ is the pressure of the gas, +- $V$ is the volume of the gas, +- $n$ is the number of moles of gas, +- $R$ is the gas constant (see [here](https://en.wikipedia.org/wiki/Gas_constant) for a table of R with various units), and +- $T$ is the temperature of the gas. + +"The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations." - Wikipedia + + + +## Sources +- [Wikipedia](https://en.wikipedia.org/wiki/Ideal_gas_law) +- [HyperPhysics](http://hyperphysics.phy-astr.gsu.edu/hbase/Kinetic/idegas.html) + +## Example + +Kevin has a 1 liter container of a mysterious (ideal) gas at 1 atm and 300 K. What is the pressure of the gas if the volume is doubled? + +There are two ways we could use the Ideal Gas Law to solve this problem: + +1. We could solve for $n = \frac{PV}{RT}$ and then use that to solve for $P_2 = \frac{nRT}{V_2}$ +2. We could notice that the right side of the equation, $nRT$ is a constant, since no moles of gas can escape, +R is a constant by definition, and the temperature is not changed in the question. Since the right side is a constant, +the left side must remain a constant. 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zvuK6^M8HfHupbMGeTd*;h?fv&-L=)aO>^1Li14?3LVCcZ z@PFZ#|BBlF>xY5f00!$z!*ul z_J26cA0hx>kkWak`G5c7*Fok#Fr^QQlDP=5eJJ^*YH-u(-1{Ga~E=npsP zzeJvU$N$3ke;ML`PxXITLz4UdJ=Op5jQs!ftPcE(B~(5vGcd07^B(Y{2vU7hA!GXS Fe*pT?L6-mk literal 0 HcmV?d00001 diff --git a/stemformulas-next/src/content/formulas/impulse-momentum-theorem.mdx b/stemformulas-next/src/content/formulas/impulse-momentum-theorem.mdx new file mode 100644 index 00000000..9659e3b8 --- /dev/null +++ b/stemformulas-next/src/content/formulas/impulse-momentum-theorem.mdx @@ -0,0 +1,23 @@ +--- +title: Impulse-Momentum Theorem +description: "The Impulse-Momentum Theorem formula." +summary: "The Impulse-Momentum Theorem formula." +tags: ["physics", "mechanics"] +date: 2025-01-04 +latex: J = \Delta p = F \Delta t +--- + +The Impulse-Momentum Theorem denotes the relationship between the change in momentum and a force that is applied over a time interval. It is given by: + +$$ J = \Delta p = F \Delta t $$ + +Where + +* $ \small \small J $ is the impulse, +* $ \small \small \Delta p $ is the change in momentum (which is how impulse is defined), +* $ \small \small F $ is the net applied force, and +* $ \small \small \Delta t $ is the time interval over which the force is applied. + +## Sources + +- [Wikipedia](https://en.wikipedia.org/wiki/Impulse_(physics)) \ No newline at end of file diff --git a/stemformulas-next/src/content/formulas/incompressible-navier-stokes.mdx b/stemformulas-next/src/content/formulas/incompressible-navier-stokes.mdx new file mode 100644 index 00000000..c5f1af67 --- /dev/null +++ b/stemformulas-next/src/content/formulas/incompressible-navier-stokes.mdx @@ -0,0 +1,62 @@ +--- +title: Incompressible Navier-Stokes Eqns. +description: "The incompressible Navier-Stokes equations describe the motion of incompressible fluid substances." +summary: "The incompressible Navier-Stokes equations describe the motion of incompressible fluid substances." +tags: ["fluid mechanics", "physics"] +date: 2025-01-11 +showTableOfContents: false +latex: \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 +--- + +The incompressible Navier-Stokes equations are a simpler form of the [Navier-Stokes equations](https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations) that describe the motion of incompressible ($ \small \rho $ = constant) fluid substances. This page focuses on the 3-dimensional equations. + +## Cartesian Coordinates +For cartesian coordinates $ \small x, y, z $ and time $ \small t $, the incompressible Navier-Stokes equations are: + +### Continuity Equation +$$ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 $$ + +### X-momentum Equation +$$ \rho \left(\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z}\right) = - \frac{\partial P}{\partial x} + \mu \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) + \rho g_x$$ + +### Y-momentum Equation +$$ \rho \left(\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z}\right) = - \frac{\partial P}{\partial y} + \mu \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 v}{\partial z^2} \right) + \rho g_y$$ + +### Z-momentum Equation +$$ \rho \left(\frac{\partial w}{\partial t} + u \frac{\partial w}{\partial x} + v \frac{\partial w}{\partial y} + w \frac{\partial w}{\partial z}\right) = - \frac{\partial P}{\partial z} + \mu \left( \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2} \right) + \rho g_z$$ + +Where + +* $\small \rho$ is the fluid density, +* $\small P$ is the pressure, +* $\small u$, $\small v$, and $\small w$ are the velocity components in the $x$, $y$, and $z$ directions, respectively. +* $\small g_x$, $\small g_y$, and $\small g_z$ are the components of the gravitational acceleration vector, and +* $\small \mu$ is the dynamic viscosity of the fluid. + +## Cylindrical Coordinates +For cylindrical coordinates $ \small r, \theta, z $ and time $ \small t $, the incompressible Navier-Stokes equations are: + +### Continuity Equation +$$ \frac{1}{r} \frac{\partial (r u_r)}{\partial r} + \frac{1}{r} \frac{\partial (u_{\theta})}{\partial \theta} + \frac{\partial (u_{z})}{\partial z} = 0 $$ + +### R-component +$$ \rho \left(\frac{\partial u_r}{\partial t} + u_r \frac{\partial u_r}{\partial r} + \frac{u_{\theta}}{r} \frac{\partial u_r}{\partial \theta} - \frac{u_{\theta}^2}{r} + u_z \frac{\partial u_r}{\partial z} \right) = - \frac{\partial P}{\partial r} + \rho g_r+ \mu \left[ \frac{1}{r} \frac{\partial}{\partial r} \left(r \frac{\partial u_r}{\partial r}\right) - \frac{u_r}{r^2} + \frac{1}{r^2} \frac{\partial^2 u_r}{\partial \theta^2} -\frac{2}{r^2} \frac{\partial u_{\theta}}{\partial \theta} + \frac{\partial^2 u_r}{\partial z^2} \right] $$ + +### $\small \theta$-component +Here is the transcribed equation in the requested single-line LaTeX format: + +$$ \rho \left(\frac{\partial u_{\theta}}{\partial t} + u_{r} \frac{\partial u_{\theta}}{\partial r} + \frac{u_{\theta}}{r} \frac{\partial u_{\theta}}{\partial \theta} + \frac{u_{r} u_{\theta}}{r} + u_{z} \frac{\partial u_{\theta}}{\partial z} \right) = - \frac{1}{r} \frac{\partial P}{\partial \theta} + \rho g_{\theta} + \mu \left[ \frac{1}{r} \frac{\partial}{\partial r} \left(r \frac{\partial u_{\theta}}{\partial r}\right) - \frac{u_{\theta}}{r^{2}} + \frac{1}{r^{2}} \frac{\partial^{2} u_{\theta}}{\partial \theta^{2}} + \frac{2}{r^{2}} \frac{\partial u_{r}}{\partial \theta} + \frac{\partial^{2} u_{\theta}}{\partial z^{2}} \right] $$ + + +### Z-component + +$$ \rho \left(\frac{\partial u_z}{\partial t} + u_r \frac{\partial u_z}{\partial r} + \frac{u_{\theta}}{r} \frac{\partial u_z}{\partial \theta} + u_z \frac{\partial u_z}{\partial z} \right) = - \frac{\partial P}{\partial z} + \rho g_z + \mu \left[ \frac{1}{r} \frac{\partial}{\partial r} \left(r \frac{\partial u_z}{\partial r}\right) + \frac{1}{r^2} \frac{\partial^2 u_z}{\partial \theta^2} + \frac{\partial^2 u_z}{\partial z^2} \right] $$ + +Where: +- $\small u_r$, $\small u_{\theta}$, and $\small u_z$ are the velocity components in the $r$, $\theta$, and $z$ directions, and +- $\small g_r$, $\small g_{\theta}$, and $\small g_z$ are the components of the gravitational acceleration vector in the $r$, $\theta$, and $z$ directions, respectively. + +## Sources + +- [Peter's Engineering Blog](http://petersengineering.blogspot.com/2015/06/navier-stokes-equation-for-3d.html) +- [Fluid Mechanics, Cengel & Cimbala 4th Ed. P474-475](https://archive.org/details/ed4_20201119) \ No newline at end of file diff --git a/stemformulas-next/src/content/formulas/inductor-energy.mdx b/stemformulas-next/src/content/formulas/inductor-energy.mdx new file mode 100644 index 00000000..2e570adb --- /dev/null +++ b/stemformulas-next/src/content/formulas/inductor-energy.mdx @@ -0,0 +1,29 @@ +--- +title: Inductor Energy +description: "Energy stored in an inductor." +summary: "Energy stored in an inductor." +tags: ["electronics", "electrical engineering"] +date: 2023-09-06 +latex: "E = \\frac{1}{2} L I^2" +--- + +## Energy +The energy stored in an inductor is given by: + +$$ E = \frac{1}{2} L I^2 $$ + +Where + +- $ \small E $ represents the energy stored in the inductor in joules (J), +- $ \small L $ is the inductance of the inductor, in henries (H), and +- $ \small I $ is the current through the inductor, in amperes (A). + + +## Sources + +- [Wikipedia](https://en.wikipedia.org/wiki/Inductor) +- [Electrical Technology](https://www.electricaltechnology.org/2020/10/inductor-inductance-formula-equations.html) + +## See also + +- [Inductor Voltage](/formulas/inductor-voltage) diff --git a/stemformulas-next/src/content/formulas/inductor-voltage.mdx b/stemformulas-next/src/content/formulas/inductor-voltage.mdx new file mode 100644 index 00000000..833e5443 --- /dev/null +++ b/stemformulas-next/src/content/formulas/inductor-voltage.mdx @@ -0,0 +1,24 @@ +--- +title: Inductor Voltage +description: "Instantaneous voltage across an inductor." +summary: "Instantaneous voltage across an inductor." +tags: ["electronics", "electrical engineering"] +date: 2023-09-06 +latex: "V = L \\frac{dI}{dt}" +--- + +The instantaneous voltage across an inductor is given by: + +$$ V = L \frac{dI}{dt} $$ + +Where + +- $ \small V $ represents the voltage across the inductor. +- $ \small L $ is the inductance of the inductor, usually measured in henries (H). +- $ \small \frac{dI}{dt} $ is the rate of change of current through the inductor with respect to time. + + +## Sources + +- [Wikipedia](https://en.wikipedia.org/wiki/Inductor) +- [Electrical Technology](https://www.electricaltechnology.org/2020/10/inductor-inductance-formula-equations.html) diff --git a/stemformulas-next/src/content/formulas/kinematic-viscosity.mdx b/stemformulas-next/src/content/formulas/kinematic-viscosity.mdx new file mode 100644 index 00000000..45b5ee6c --- /dev/null +++ b/stemformulas-next/src/content/formulas/kinematic-viscosity.mdx @@ -0,0 +1,22 @@ +--- +title: Kinematic Viscosity +description: "The ratio of the dynamic viscosity to the density of a fluid." +summary: "The ratio of the dynamic viscosity to the density of a fluid." +tags: ["fluid mechanics", "physics"] +date: 2024-03-13 +latex: "\nu = \\frac{\\mu}{\\rho}" +--- + +Kinematic viscosity is the ratio of the viscosity (dynamic viscosity) to the density of a fluid: + +$$ \nu = \frac{\mu}{\rho} $$ + +Where: + +- \$ \small \nu\ $ is the kinematic viscosity of the fluid, +- \$ \small \mu\ $ is the dynamic viscosity of the fluid, and +- \$ \small \rho\ $ is the density of the fluid. + +## Sources + +- [Wikipedia](https://en.wikipedia.org/wiki/Viscosity#Kinematic_viscosity) diff --git a/stemformulas-next/src/content/formulas/kinetic-energy.mdx b/stemformulas-next/src/content/formulas/kinetic-energy.mdx new file mode 100644 index 00000000..157528fa --- /dev/null +++ b/stemformulas-next/src/content/formulas/kinetic-energy.mdx @@ -0,0 +1,30 @@ +--- +title: "Kinetic Energy" +description: "The energy of motion." +summary: "The energy of motion." +tags: ["physics"] +date: 2022-12-24 +latex: E_k = \frac{1}{2}mv^2 +--- + +$$ E_k = \frac{1}{2}mv^2 $$ + +Where +* $E_k$ is the kinetic energy of the object, +* $m$ is the mass of the object, and +* $v$ is the velocity of the object. + +## Sources +- [Wikipedia](https://en.wikipedia.org/wiki/Kinetic_energy) +- [Brittanica](https://www.britannica.com/science/kinetic-energy) + +## Example +Kevin throws a 400,000 kg spaceship straight down off of an infinitely tall cliff. After some time, it reaches a terminal velocity of 500 m/s. What is the kinetic energy of the spaceship at this point? + +We know that kinetic energy is equal to $\small \frac{1}{2}mv^2$, so we can plug in the values to get the answer: + +$$ \scriptsize E_k = \frac{1}{2}mv^2 \Longrightarrow E_k = \frac{1}{2} \cdot 400000 \cdot 500^2 \Longrightarrow E_k = 5.0 \cdot 10^{10} J $$ + + +## See also +- [Potential Energy](/formulas/gravitational-potential-energy/) diff --git a/stemformulas-next/src/content/formulas/laplaces-equation.mdx b/stemformulas-next/src/content/formulas/laplaces-equation.mdx new file mode 100644 index 00000000..f1888afe --- /dev/null +++ b/stemformulas-next/src/content/formulas/laplaces-equation.mdx @@ -0,0 +1,34 @@ +--- +title: "Laplace's Equation" +description: "Laplace's equation is a common partial differential equation" +summary: "Laplace's equation is a common partial differential equation" +tags: ["math", "differential equations", "physics"] +date: 2022-12-24 +latex: \nabla^2 f = \nabla \cdot \nabla f = 0 +--- + +$$ \nabla^2 f = \nabla \cdot \nabla f = 0 $$ + +Where +* $f$ is a twice-differentiable function, +* $\nabla$ is the [gradient operator](https://en.wikipedia.org/wiki/Gradient), +* $\nabla^2$ is the [Laplace operator](https://en.wikipedia.org/wiki/Laplace_operator), and +* $\nabla \cdot $ is the [divergence operator](https://en.wikipedia.org/wiki/Divergence). + +## In Different Coordinate Systems + +#### In rectangular coordinates: + +$$ \scriptsize \nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} = 0$$ + +#### In cylindrical coordinates: +$$ \scriptsize \nabla^2 f = \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial f}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 f}{\partial \phi^2} + \frac{\partial^2 f}{\partial z^2} = 0$$ + +#### In spherical coordinates: +$$ \scriptsize \nabla^2 f = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial f}{\partial r} \right) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial f}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta}\frac{\partial^2 f}{\partial \varphi^2} = 0$$ + +## Sources +- [Wikipedia](https://en.wikipedia.org/wiki/Laplace%27s_equation) +- Griffith's Introduction to Electrodynamics' back cover + + diff --git a/stemformulas-next/src/content/formulas/lhopitals-rule.mdx b/stemformulas-next/src/content/formulas/lhopitals-rule.mdx new file mode 100644 index 00000000..acabe4b6 --- /dev/null +++ b/stemformulas-next/src/content/formulas/lhopitals-rule.mdx @@ -0,0 +1,33 @@ +--- +title: L'Hopital's Rule +description: L'Hopital's rule is a method for evaluating limits of indeterminate forms. +summary: L'Hopital's rule is a method for evaluating limits of indeterminate forms. +tags: ["math", "calculus", "limits"] +date: 2023-01-03 +latex: \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} +--- + +L'hopital's rule is a method for evaluating limits of [indeterminate form](https://en.wikipedia.org/wiki/Indeterminate_form). + +$$\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$$ + +Where +- $ \small f(x) $ and $ \small g(x) $ are functions, +- $\small '$ denotes the derivative of a function, and +- $ \small c $ is the point at which the limit is being evaluated + +## Sources +- [Wikipedia](https://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule) +- [Paul's Online Notes](https://tutorial.math.lamar.edu/classes/calci/lhospitalsrule.aspx) + +## Example + +Calculate the limit of $\small f(x) = \frac{x^2 - 9}{x - 3}$ as $\small x$ approaches $\small 3$. + +Since plugging in 3 yields 0/0 which is an indeterminate form, we can apply L'Hopital's rule, with $ \small f(x) = x^2 - 9$ and $ \small g(x) = x - 3$. + +Taking both derivatives, we get $ \small f'(x) = 2x$ and $ \small g'(x) = 1$. + +Plugging into the right side of the L'Hopital's rule, we get that + +$$ \lim_{x \to 3} \frac{f(x)}{g(x)} = \lim_{x \to 3} \frac{f'(x)}{g'(x)} = \lim_{x \to 3} \frac{2x}{1} = 6$$ diff --git a/stemformulas-next/src/content/formulas/light-wavelength-frequency.mdx b/stemformulas-next/src/content/formulas/light-wavelength-frequency.mdx new file mode 100644 index 00000000..1de09ba5 --- /dev/null +++ b/stemformulas-next/src/content/formulas/light-wavelength-frequency.mdx @@ -0,0 +1,32 @@ +--- +title: "Light Wavelength and Frequency Relationship" +description: "The relationship between the wavelength and frequency of light." +summary: "The relationship between the wavelength and frequency of light." +tags: ["physics", "optics"] +date: 2023-03-24 +latex: c = \lambda f +--- + +The relationship between the wavelength and frequency of light in a vacuum is: + +$$ c_0 = \lambda f $$ + +Where + +* $\scriptsize \lambda$ is the wavelength of the light, +* $\scriptsize c_0$ is the speed of light in a vacuum, and +* $\scriptsize f$ is the frequency of the light. + +For a medium with a refractive index of $\scriptsize n$, the relationship is: + +$$ \frac{c_0}{n} = \frac{\lambda}{n} f $$ + +That is to say, the speed of light and wavelength change in a medium, but the frequency remains the same. + +## Sources + +- [Wikipedia](https://en.wikipedia.org/wiki/Wavelength) +- [Brittanica](https://www.britannica.com/science/wavelength) + +## See also +- [Energy of light](/formulas/energy-of-light/) diff --git a/stemformulas-next/src/content/formulas/linear-thermal-expansion.mdx b/stemformulas-next/src/content/formulas/linear-thermal-expansion.mdx new file mode 100644 index 00000000..5a11afed --- /dev/null +++ b/stemformulas-next/src/content/formulas/linear-thermal-expansion.mdx @@ -0,0 +1,24 @@ +--- +title: Thermal expansion +description: "The thermal expansion formula." +summary: "The thermal expansion formula." +tags: ["physics", "material engineering"] +date: 2023-07-03 +latex: \Delta L = L_0 \alpha \Delta T +--- + +The thermal expansion formula is: + +$$ \Delta L = L_0 \alpha \Delta T $$ + +Where + +* $\small \Delta L$ is the change in length, +* $\small L_0$ is the initial length, +* $\small \alpha$ is the coefficient of linear expansion, and +* $\small \Delta T$ is the change in temperature in Kelvin. + +## Sources + +- [Wikipedia](https://en.wikipedia.org/wiki/Thermal_expansion) +- [Engineering Toolbox](https://www.engineeringtoolbox.com/linear-thermal-expansion-d_1379.html) \ No newline at end of file diff --git a/stemformulas-next/src/content/formulas/log-laws.mdx b/stemformulas-next/src/content/formulas/log-laws.mdx new file mode 100644 index 00000000..177b8eaf --- /dev/null +++ b/stemformulas-next/src/content/formulas/log-laws.mdx @@ -0,0 +1,34 @@ +--- +title: Log Laws +description: "Properties of logarithms." +summary: "Properties of logarithms." +tags: ["mathematics"] +date: 2023-09-13 +latex: \log_b(a) = \frac{\log_c(a)}{\log_c(b)} +--- + +Logarithms are the inverse of exponentials. They are useful for solving equations with exponents, and they are used in many other areas of mathematics. Here are some commonly used logarithm properties. + + +For any positive real number $ \small b \neq 1 $, and positive real numbers $ \small a $, and $ \small c $, the following properties hold: + +$$ \log_b(1) = 0 $$ +$$ \log_b(b) = 1 $$ +$$ \log_b(a \cdot c) = \log_b(a) + \log_b(c) $$ +$$ \log_b \left(\frac{a}{c} \right) = \log_b(a) - \log_b(c) $$ +$$ \log_b(a^c) = c \cdot \log_b(a) $$ +$$ b^{\log_b(k)} = k $$ + +Note that $ k $ can be any real number in the last equation. + +## Change of Base Formula + +The change of base formula is useful for converting logarithms to different bases. + +$$ \log_b(a) = \frac{\log_c(a)}{\log_c(b)} $$ + + +## Sources + +- [ChiliMath](https://www.chilimath.com/lessons/advanced-algebra/logarithm-rules/) +- [Wikipedia](https://en.wikipedia.org/wiki/List_of_logarithmic_identities) diff --git a/stemformulas-next/src/content/formulas/lorentz-transformations.mdx b/stemformulas-next/src/content/formulas/lorentz-transformations.mdx new file mode 100644 index 00000000..363953f5 --- /dev/null +++ b/stemformulas-next/src/content/formulas/lorentz-transformations.mdx @@ -0,0 +1,22 @@ +--- +title: Lorentz Transformations +description: Lorentz transformations are used to transform between inertial frames of reference. +summary: Lorentz transformations are used to transform between inertial frames of reference. +tags: ["physics", "relativity"] +date: 2023-01-02 +latex: x' = \frac{x-vt}{\sqrt{1-v^2/c^2}} +--- + +For a fixed frame of reference $\small x, y, z, t$, the following transformations can be used to transform to a frame of reference moving with velocity $\small v$ with respect to the fixed frame: + +$$\small x' = \frac{x-vt}{\sqrt{1-v^2/c^2}}$$ +$$ \small y' = y$$ +$$ \small z' = z$$ +$$ \small t' = \frac{t-vx/c^2}{\sqrt{1-v^2/c^2}}$$ + +Where +* $c$ is the speed of light + +## Sources +- [Wikipedia](https://en.wikipedia.org/wiki/Lorentz_transformation) +- [HyperPhysics](http://hyperphysics.phy-astr.gsu.edu/hbase/Relativ/ltrans.html) diff --git a/stemformulas-next/src/content/formulas/matrix-multiplication.mdx b/stemformulas-next/src/content/formulas/matrix-multiplication.mdx new file mode 100644 index 00000000..ffd3d316 --- /dev/null +++ b/stemformulas-next/src/content/formulas/matrix-multiplication.mdx @@ -0,0 +1,20 @@ +--- +title: Matrix Multiplication +description: "How to multiply matrices." +summary: "How to multiply matrices." +tags: ["math", "linear algebra"] +date: 2023-07-08 +latex: \scriptsize \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} w & x \\ y & z \end{pmatrix} = \begin{pmatrix} aw + by & ax + bz \\ cw + dy & cx + dz \end{pmatrix} +--- + +In general for, any matrix multiplication, the ith row and jth column of the product matrix is the [dot product](/formulas/dot-product) of the ith row of the first matrix and the jth column of the second matrix. + +For example, for two 2x2 matrices, the product matrix is: + +$$ \begin{pmatrix} a & b \\\\ c & d \end{pmatrix} \begin{pmatrix} w & x \\\\ y & z \end{pmatrix} = \begin{pmatrix} aw + by & ax + bz \\\\ cw + dy & cx + dz \end{pmatrix}$$ + + +## Sources + +- [Math is Fun](https://www.mathsisfun.com/algebra/matrix-multiplying.html) +- [Khan Academy](https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:matrices/x9e81a4f98389efdf:multiplying-matrices-by-matrices/a/multiplying-matrices) diff --git a/stemformulas-next/src/content/formulas/maxwells-equations.mdx b/stemformulas-next/src/content/formulas/maxwells-equations.mdx new file mode 100644 index 00000000..0eaeb4fe --- /dev/null +++ b/stemformulas-next/src/content/formulas/maxwells-equations.mdx @@ -0,0 +1,78 @@ +--- +title: Maxwell's Equations +description: "The four Maxwell's equations describing electromagnetism." +summary: "The four Maxwell's equations describing electromagnetism." +tags: ["physics", "electromagnetism"] +date: 2023-08-02 +latex: \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0},\ \nabla \cdot \mathbf{B} = 0,... +--- + +Maxwell's Equations represent the fundamental laws of electromagnetism. + +There are two common forms of Maxwell's Equations: the differential form and the integral form. +## Differential form + +In differential form, Maxwell's equations are given by: + +$$ \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon\_0} $$ + +$$ \nabla \cdot \mathbf{B} = 0 $$ + +$$ \nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}$$ + +$$ \nabla \times \mathbf{B} = \mu\_0 \mathbf{J} + \mu\_0 \varepsilon\_0 \frac{\partial \mathbf{E}}{\partial t} $$ + +Where: + +* $\small \mathbf{E} $ is the electric field vector, +* $\small \mathbf{B} $ is the magnetic field vector, +* $\small \rho $ is the electric charge density, +* $\small \varepsilon_0 $ is the vacuum permittivity (electric constant), +* $\small \mu_0 $ is the vacuum permeability (magnetic constant), and +* $\small \mathbf{J} $ is the electric current density. + +Respectively, these equations are Gauss's Law, Gauss's Law for Magnetism, Faraday's Law, and Ampère's Law with Maxwell's Addition. + +In a vacuum (where there is no charge or current), these simplify to: + +$$ \nabla \cdot \mathbf{E} = 0 $$ +$$ \nabla \cdot \mathbf{B} = 0 $$ +$$ \nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}$$ +$$ \nabla \times \mathbf{B} = \mu\_0 \varepsilon\_0 \frac{\partial \mathbf{E}}{\partial t} $$ + + +## Integral form + +In integral form, Maxwell's equations are given by: + +$$ \oiint\_{\partial \Omega} \mathbf{E} \cdot d\mathbf{S} = \frac{Q}{\varepsilon\_0} = \frac{1}{\varepsilon\_0}\iiint_{\Omega} \rho \ dV $$ +$$ \oiint\_{\partial \Omega} \mathbf{B} \cdot d\mathbf{S} = 0 $$ +$$ \oint\_{\partial \Sigma} \mathbf{E} \cdot d\mathbf{l} = - \frac{\text{d}}{\text{d} t} \iint\_{\Sigma} \mathbf{B} \cdot d\mathbf{S} $$ +$$ \oint\_{\partial \Sigma} \mathbf{B} \cdot d\mathbf{l} = \mu\_0 \iint\_{\Sigma} \mathbf{J} \cdot d\mathbf{S} + \mu\_0 \varepsilon\_0 \frac{\text{d}}{\text{d} t} \iint\_{\Sigma} \mathbf{E} \cdot d\mathbf{S} $$ + +Where: + +* $ \small \mathbf{E} $ is the electric field vector, +* $\small \mathbf{B} $ is the magnetic field vector, +* $\small Q $ is the electric charge enclosed by the volume $\small \Omega $, +* $ \mathbf{J} $ is the electric current density, +* $ \oiint\_{\partial \Omega} $ is the surface integral over the closed surface $ \partial \Omega $, +* $ \iiint\_{\Omega} $ is the volume integral over the volume $\small \Omega $, +* $ \rho $ is the electric charge density, +* $ \oint\_{\partial \Sigma} $ is the line integral over the closed curve $ \partial \Sigma $, +* $ \iint\_{\Sigma} $ is the surface integral over the surface $\small \Sigma $, +* $ \varepsilon_0 $ is the vacuum permittivity (electric constant), +* $ \mu_0 $ is the vacuum permeability (magnetic constant), and +* $ d \mathbf{l} $, $ d \mathbf{S} $, and $ dV $ are the line, surface, and volume elements, respectively. + +In a vacuum (where there is no charge or current), these simplify to: + +$$ \oiint\_{\partial \Omega} \mathbf{E} \cdot d\mathbf{S} = 0 $$ +$$ \oiint\_{\partial \Omega} \mathbf{B} \cdot d\mathbf{S} = 0 $$ +$$ \oint\_{\partial \Sigma} \mathbf{E} \cdot d\mathbf{l} = - \frac{\text{d}}{\text{d} t} \iint\_{\Sigma} \mathbf{B} \cdot d\mathbf{S} $$ +$$ \oint\_{\partial \Sigma} \mathbf{B} \cdot d\mathbf{l} = \mu\_0 \varepsilon\_0 \frac{\text{d}}{\text{d} t} \iint\_{\Sigma} \mathbf{E} \cdot d\mathbf{S} $$ + +## Sources + +- [Wikipedia](https://en.wikipedia.org/wiki/Maxwell's_equations#Formulation_in_SI_units_convention) +- [HyperPhysics](http://hyperphysics.phy-astr.gsu.edu/hbase/electric/maxeq.html) \ No newline at end of file diff --git a/stemformulas-next/src/content/formulas/modulus-of-resilience.mdx b/stemformulas-next/src/content/formulas/modulus-of-resilience.mdx new file mode 100644 index 00000000..5a8734e4 --- /dev/null +++ b/stemformulas-next/src/content/formulas/modulus-of-resilience.mdx @@ -0,0 +1,32 @@ +--- +title: Modulus of Resilience +description: "The formula for the modulus of resilience." +summary: "The formula for the modulus of resilience." +tags: ["physics", "material engineering"] +date: 2023-07-03 +latex: U_r = \int_{0}^{\epsilon_Y}\sigma\mathop{d\epsilon} \approx \frac{\sigma_{YS}^2}{2E} +--- + +The modulus of resilience is given by: + +$$ U_r = \int_{0}^{\epsilon_Y}\sigma\mathop{d\epsilon} \approx \frac{\sigma_{YS}^2}{2E} $$ + +Where + +* $ \small U_r $ is the modulus of resilience, +* $ \small \sigma $ is the stress, +* $ \small \epsilon $ is the strain, +* $ \small \epsilon_Y $ is the yield strain, +* $ \small \sigma_{YS} $ is the yield strength, and +* $ \small E $ is the elastic modulus (Young's modulus). + +Note that the first formula is th exact formula, equal to the area under the stress-strain curve up to the yield strain, and the second formula is an approximation for the modulus of resilience, which is valid for materials with a linear elastic region. + +## Sources + +- [Mechical](https://www.mechical.com/2022/07/modulus-of-resilience.html) +- [Mechanicalc](https://mechanicalc.com/reference/mechanical-properties-of-materials#modulus-of-resilience) + +## See also +- [Hooke's Law](/formulas/hookes-law-stress-strain/) +- [Modulus of Toughness](/formulas/modulus-of-toughness/) diff --git a/stemformulas-next/src/content/formulas/modulus-of-toughness.mdx b/stemformulas-next/src/content/formulas/modulus-of-toughness.mdx new file mode 100644 index 00000000..dd58d99b --- /dev/null +++ b/stemformulas-next/src/content/formulas/modulus-of-toughness.mdx @@ -0,0 +1,30 @@ +--- +title: Modulus of Toughness +description: "The formula for the modulus of toughness." +summary: "The formula for the modulus of toughness." +tags: ["physics", "material engineering"] +date: 2023-07-03 +latex: U_t = \int_{0}^{\epsilon_f}\sigma\mathop{d\epsilon} +--- + +The modulus of toughness is given by: + +$$ U_t = \int_{0}^{\epsilon_f}\sigma\mathop{d\epsilon} $$ + +Where + +* $ \small U_t $ is the modulus of toughness, +* $ \small \sigma $ is the stress, +* $ \small \epsilon $ is the strain, and +* $ \small \epsilon_f $ is the strain at fracture. + +Note that this is the area under the stress-strain curve up to the failure/fracture strain. See the sources for some geometric approximations for the modulus of toughness. + +## Sources + +- [Mechanicalc](https://mechanicalc.com/reference/mechanical-properties-of-materials#modulus-of-toughness) +- [Wikipedia](https://en.wikipedia.org/wiki/Toughness) + +## See also +- [Hooke's Law](/formulas/hookes-law-stress-strain/) +- [Modulus of Resilience](/formulas/modulus-of-resilience/) \ No newline at end of file diff --git a/stemformulas-next/src/content/formulas/newtonian-momentum.mdx b/stemformulas-next/src/content/formulas/newtonian-momentum.mdx new file mode 100644 index 00000000..13281c6d --- /dev/null +++ b/stemformulas-next/src/content/formulas/newtonian-momentum.mdx @@ -0,0 +1,27 @@ +--- +title: "Momentum (Newtonian)" +description: "The momentum of a body in Newtonian mechanics" +summary: "The momentum of a body in Newtonian mechanics" +tags: ["physics", "mass", "velocity"] +date: 2022-12-08 +latex: p = mv +--- +$$p = mv$$ + +Where +* $p$ is the momentum of the body, +* $m$ is the mass of the body, and +* $v$ is the velocity of the body + +This formula relates the momentum of a body to its mass and velocity. + +## Sources +- [Wikipedia](https://en.wikipedia.org/wiki/Momentum) +- [Hyperphysics](http://hyperphysics.phy-astr.gsu.edu/hbase/mom.html) + +## Example +Kevin throws a baseball with a mass of 0.5 kg at a speed of 5000000 m/s. What is the momentum of the baseball? + +We know that momentum is equal to mass times velocity, so we can plug in the values to get the answer. + +$$p = mv \Longrightarrow p = 0.5 \cdot 5000000 \Longrightarrow p = 2500000 \frac{kg \cdot m}{s}$$ diff --git a/stemformulas-next/src/content/formulas/newtons-2nd-law.mdx b/stemformulas-next/src/content/formulas/newtons-2nd-law.mdx new file mode 100644 index 00000000..4e4a06a6 --- /dev/null +++ b/stemformulas-next/src/content/formulas/newtons-2nd-law.mdx @@ -0,0 +1,26 @@ +--- +title: "Newton's 2nd Law" +description: "The law that describes the acceleration of an object in Newtonian mechanics." +summary: "The law that describes the acceleration of an object in Newtonian mechanics." +tags: ["physics", "mechanics"] +date: 2022-12-04 +latex: F = ma +--- +$$F = ma$$ + +Where +$F$ is the net force being applied to the object, +$m$ is the mass of the object, and +$a$ is the acceleration of the object. + +Newton's second law is a statement of Newtonian mechanics that the net force on a body is equal to the body's mass times its acceleration. + +## Sources +- [Wikipedia](https://en.wikipedia.org/wiki/Newton%27s_laws_of_motion) +- [Brittanica](https://www.britannica.com/science/Newtons-laws-of-motion/Newtons-second-law-F-ma) + +## Example +Kevin pushes his 10 kg brain across the floor with a force of 100 N. Friction pushes back against his brain statically +with 10 N of force. What is the acceleration of the brain? + +$$ F = ma \Longrightarrow a = \frac{F}{m} = \frac{100N - 10N}{10kg} = 9.8 m/s^2 $$ diff --git a/stemformulas-next/src/content/formulas/newtons-law-of-cooling.mdx b/stemformulas-next/src/content/formulas/newtons-law-of-cooling.mdx new file mode 100644 index 00000000..ad63f944 --- /dev/null +++ b/stemformulas-next/src/content/formulas/newtons-law-of-cooling.mdx @@ -0,0 +1,26 @@ +--- +title: Newton's Law of Cooling +description: "The formula describing Newton's Law of Cooling." +summary: "Newton's Law of Cooling." +tags: ["physics", "thermodynamics"] +date: 2023-08-01 +latex: \frac{dT}{dt} = -k(T - T_{\text{env}}) +--- + +Newton's law of Cooling says that the rate of change of temperature of an object is proportional to the difference between the object's temperature and the ambient temperature. + +It is given by: + +$$ \frac{dT}{dt} = -k(T - T_{\text{env}}) $$ + +Where + +* $ \frac{dT}{dt} $ is the rate of change of temperature with respect to time, +* $ T $ is the temperature of the object at time $ t $, +* $ T_{\text{env}} $ is the ambient (surrounding) temperature, and +* $ k $ is the cooling rate constant. + +## Sources + +- [BYJU'S](https://byjus.com/jee/newtons-law-of-cooling/) +- [Wikipedia](https://en.wikipedia.org/wiki/Newton%27s_law_of_cooling) diff --git a/stemformulas-next/src/content/formulas/ohms-law.mdx b/stemformulas-next/src/content/formulas/ohms-law.mdx new file mode 100644 index 00000000..1f4d7879 --- /dev/null +++ b/stemformulas-next/src/content/formulas/ohms-law.mdx @@ -0,0 +1,31 @@ +--- +title: "Ohm's Law" +description: "Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points." +summary: "Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points." +tags: ["physics", "electrical engineering"] +date: 2022-11-27 +latex: V = IR +--- + +Ohm's law is given by: + +$$V = IR$$ + +Where +$V$ is the voltage across the conductor (e.g. a wire), +$I$ is the current through the conductor, and +$R$ is the resistance of the conductor. + +Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. - Wikipedia + +## Sources +- [Wikipedia](https://en.wikipedia.org/wiki/Ohm%27s_law) +- [HyperPhysics](http://hyperphysics.phy-astr.gsu.edu/hbase/electric/ohmlaw.html) + +## Example + +Kevin measures the current through a 10 ohm resistor to be 0.5 A. He measures the current with the resistor actually connected to the circuit (sorry Mr. Sidney). What is the voltage across the resistor? + +$$ V = IR \Longrightarrow V = 10 * 0.5 = 5V $$ + + diff --git a/stemformulas-next/src/content/formulas/pendulum-frequency-period.mdx b/stemformulas-next/src/content/formulas/pendulum-frequency-period.mdx new file mode 100644 index 00000000..4760407b --- /dev/null +++ b/stemformulas-next/src/content/formulas/pendulum-frequency-period.mdx @@ -0,0 +1,32 @@ +--- +title: Pendulum Period/Frequency +description: "The formula for the period and frequency of a simple pendulum." +summary: "The formula for the period of a simple pendulum." +tags: ["physics", "mechanics"] +date: 2025-01-04 +latex: T_p = 2 \pi \sqrt{\frac{\ell}{g}} +--- + +The formula for the period of a simple pendulum is: + +$$ T_p = 2 \pi \sqrt{\frac{\ell}{g}} $$ + +Where + +* $\small T_p$ is the period of oscillation, +* $\small \ell$ is the length of the pendulum, and +* $\small g$ is the acceleration due to gravity. + +The frequency of oscillation is the reciprocal of the period: + +$$ f_p = \frac{1}{2 \pi} \sqrt{\frac{g}{\ell}} $$ + +Where + +* $\small f_p$ is the frequency of oscillation. + +**Note:** These formulas use an approximation that assume small oscillations (small angles) and neglects air resistance, damping, and the mass of the string. + +## Sources + +- [LibreTexts Physics](https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_(OpenStax)/Book%3A_University_Physics_I_-_Mechanics_Sound_Oscillations_and_Waves_(OpenStax)/15%3A_Oscillations/15.05%3A_Pendulums) \ No newline at end of file diff --git a/stemformulas-next/src/content/formulas/plane-equation.mdx b/stemformulas-next/src/content/formulas/plane-equation.mdx new file mode 100644 index 00000000..65659a76 --- /dev/null +++ b/stemformulas-next/src/content/formulas/plane-equation.mdx @@ -0,0 +1,42 @@ +--- +title: Equation of a Plane +description: "The equation of a plane formula." +summary: "The equation of a plane formula." +tags: ["geometry", "math"] +date: 2023-07-04 +latex: ax + by + cz = d +--- + +## Scalar Equation + +The scalar equation of a plane is: + +$$ ax + by + cz = d $$ + +Where + +* $ \small a, b, c $ are the components of the normal vector to the plane, and +* $ \small \small d $ is the smallest distance from the origin to the plane. + + +## Vector Equation + +The vector equation of a plane is: + +$$ (r - r_0 )\cdot n = 0 $$ + +Where + +* $ \small r $ represents a point on the plane, +* $ \small r_0 $ is a specific point on the plane (the position vector of a known point), +* $ \small n $ is the normal vector to the plane. + +What this equation says is that the plane is defined by all the position vectors that start on the plane that are perpendicular to the normal vector. + +The vector equation for the plane can be converted to the scalar equation by expanding the vectors $ \small n = (a, b, c)$, $ r = (x, y, z) $, and $ r_0 = (x_0, y_0, z_0) $ and then taking the dot product and grouping the constant terms into $ \small d $. + + +## Sources + +- [Paul's Online Math Notes](https://tutorial.math.lamar.edu/classes/calciii/eqnsofplanes.aspx) +- [Wikipedia](https://en.wikipedia.org/wiki/Euclidean_planes_in_three-dimensional_space#Point%E2%80%93normal_form_and_general_form_of_the_equation_of_a_plane) \ No newline at end of file diff --git a/stemformulas-next/src/content/formulas/poissons-ratio.mdx b/stemformulas-next/src/content/formulas/poissons-ratio.mdx new file mode 100644 index 00000000..ea20f882 --- /dev/null +++ b/stemformulas-next/src/content/formulas/poissons-ratio.mdx @@ -0,0 +1,24 @@ +--- +title: Poisson's Ratio +description: "The Poisson's Ratio formula." +summary: "The Poisson's Ratio formula." +tags: ["physics", "material engineering"] +date: 2023-07-03 +latex: \nu = - \frac{\epsilon_x}{\epsilon_z} = -\frac{\epsilon_y}{\epsilon_z} +--- + +Poisson's Ratio is given by: + +$$ \nu = - \frac{\epsilon_x}{\epsilon_z} = -\frac{\epsilon_y}{\epsilon_z} $$ + +Where + +* $\small \nu $ is Poisson's Ratio, +* $\small \epsilon_x $ is the strain in the x-direction, +* $\small \epsilon_y $ is the strain in the y-direction, and +* $\small \epsilon_z $ is the strain in the z-direction. + +## Sources + +- [Wikipedia](https://en.wikipedia.org/wiki/Poisson%27s_ratio) +- [BYJU'S](https://byjus.com/physics/poissons-ratio/) \ No newline at end of file diff --git a/stemformulas-next/src/content/formulas/preview.png b/stemformulas-next/src/content/formulas/preview.png new file mode 100644 index 0000000000000000000000000000000000000000..a746aafdf44fa15a728c5d66248eb9a5b4346142 GIT binary patch literal 52972 zcmeFZhds8XRN6414_ud@YD`an_kd^F}gzS)&Y!c!mduJs(n{0m9 zQN6#vKjL$4-Cj5KI?j2H$K$&0>v{$&D@qaKQQ{#G2tpZYaa9BY`#S=0RvY&${L4;v zsS^V6tyD%_Ox^v}uQ4kJy}`s=hfeHluKsjlB=ij_bc`V}rmP=W+Nx_>AH5nXPI-IT z;t`jOfLgzBN?u}a?(02{)T}IFb&t%kMM){-Pgx zvHjoAq-XA3{rmId>vxgo{{F0tg=_ZjEg^_Bg1pK<+ri^OB)e_!K_Vr?4YLyADWH&kXkt_RE7${ z*qt*eisa#vw-dkKsap>}*-AaWY{C+cwkp@9w?hmv{4MaA5^-kMX9T&pHnK&J7AoqG zmWN9%dKbzRvoq2Jla38*w>*7V$k20}O-)T@*th!-g=T*jzO0-`>YydJKV4W@*x`DF z-{H?_rIKM5ld$7}d|ehY=8p)(AQ$OfOxr}nkXaw(AZ58Jsi~;GnwRE;4tsMY5*XZ# zS9)x7mFZ76lEvSCaS7``lHg}e4^&8R%0!t}=H|A|I4(F>sWaWgJsXvjf9YtCRHE{D zf5{48LF3$Go6CRt+hbL+4r9xh*V6P13=GD_3F*iZmgJ6E!Ih8G!M9$&*JepmmX4x- z#-(*bg|u||9SgdnqoWMlPZqxWe;;sLM_lEtBv@i`W)do8x~vZy({{L!;!u`+%5@dK^4uR_aaU&YCWmjmpMwKObX!-1VsFAeFxa1bR4+n8@LIIl;!Sek42Q)3LLJhpov4ky!=WKg!Xr+SN5R?S@M-^YTJ`#WUf`GCciv zWpVv4hD)Bu!RcAxepsx{Vqjn(nZcRbc^h+8lZpObwCfpV5-Kr992B@%?xBh8?d@4v z3@W6Xdo}4u%x>qsnmcs<^aLE1(+?%0=p|GbZry5aZOuhWk~Wu=lMo~jgSmQ!9}@XzjFGS5>0YOT!~?$ z%QV-nWspi{_qsScbMx?Y$J~_HKZ_WO`FA4-iOqTzwz|5yU|SsJ^e*V?=`AfSEl7!7 ze2GRNY(@UorI299M18h9+}zxpoQ>6A%RYP%NKsS=P;6`b?;Y&0WeEutl$3gzn5<^qB!5=A~r`Cd3@S zGF1&$VAUZ5$%PR%+78xK15}e|OR&Wn>02q;pO!h>Iww zX~ywtxa6O0WFGMELy$3-uo_o);FaBLHDaizH##toHMkV&=?u+frE8uXbHHzKNhPLw zDy1Fi+%yJ703o|n%vU|HXhE7-hyMIlHs%)!gYs;DLqH>X!fu5dC=`e*!rT&Wt z`y_~l*}vaul%aAVj9w-rJuEyt;NE}Zy@STZ>RMVh<+?2i0=Z}4@X!A_{02!_V;7%6 zA$#)V30zaPMCdHAI9yy@iit16^*sA`Ev6^FeoaF`fg9Acl`(8GJT- z-@jEe!{~4|^UG^%YqPQf3j^Vz89CkmyXmO>>BKX-BO@c71v)6yNSlp|on2ky;^WmA z3E{Z@cM}>Udo8)0;Sz5Ka)lMcM)~$lu|P8!JwqxXhFLrv3QT{wn7)&^iaO~{lHgtM zI(@cVxc=$LD0&SoEiFw=E{)#A^f$Hn5l;qxy{mH%npi$`(r>CIwQ)lrw6OjjdCSyh zRLQWdpWn&+Jn~_mC3kFW?DqC{X4ellgNO%DGcMAzBb3|!?hRWi7v%K1JEk~>(#t@5 z3l9&MATZn5t6@R=(@FPyYQoZq%Po{5l~5s#_jnnS{spRjNjlP>&XPL`t%z=)aV%lk ztkv(%NJHXH+%S=kdYkEX3+*V9I@ju&S$z(nP58G8OT0wSK%bG6l46YMK5ce{goF|@ zSnl+V$z!HmiSi6h(U-y{Z^%SJOI{acpcPvj%iDMZUJ6dlYk-*)rww_OXWuxT^px_|$EMn;BuS6aXI=*S4= zwQG!Y+P>87NnhqR!#G3kkxH!ZdBb}Qi|!=KFK&B=Hp6DMD;?I=)rC7W&j4$7!c?qA zKl2T@IlLSlxhTWGeftInsLjHFE@?B(gDU&(U70q^l`B_Vt5Z;-$sHS%ahaX|#^fO{ zcU!bsczJorg2j!UEMu9IJ2ash>*!dO4$CuKt`{lSHdN#%Ml>w`Z7f54eZ5*2pUE@a z77?L|XtjbZ%NCDba4rci=zBB&tR&Wm!+n2iq12*NQBiUK==d((OW90kffP8+lo}p4PDl8x%z;EH{ zDFFA2B-oTI7a7uKTJkOJOKw(H7VMAw{G}4ieYpR(V)8y0cz7YGDLOjZySmZH^5MhG z%=nxt)20_8M2sZCHNQt}I)ygBPBjHBAS|

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proj_{\vec{b}}(\vec{a})=\frac{\vec{a}\cdot\vec{b}}{\|\vec{b}\|^2}\vec{b}=(\vec{a}\cdot\hat{b})\hat{b} +--- + +For any vectors $\small \vec{a}$ and $\small \vec{b}$: +![diagram-dont-css](projection.png) + +The projected (red) and perpendicular (green) vector can be calculated as follows: + +$$ proj_{\vec{b}}(\vec{a})=\frac{\vec{a}\cdot\vec{b}}{\|\vec{b}\|^2}\vec{b}=(\vec{a}\cdot\hat{b})\hat{b} $$ + +$$ perp_{\vec{b}}(\vec{a})=\vec{a}-proj_{\vec{b}}(\vec{a}) $$ + +Where +* $\small \vec{a}$ is the vector to be projected +* $\small \vec{b}$ is the vector to be projected onto +* $\cdot$ is the [dot product](/formulas/dot-product/) +* $\small \|\vec{b}\|^2$ is the [magnitude](https://en.wikipedia.org/wiki/Magnitude_(mathematics)#Euclidean_vector_space) of $\small \vec{b}$, squared +* $\small \hat{b}$ is the [unit vector](/formulas/unit-vector/) in the direction of $\small \vec{b}$ + + +## Sources +- [Wikipedia](https://en.wikipedia.org/wiki/Vector_projection) +- [The Organic Chemistry 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"sin", "cos", "tan", "math", "identity"] +date: 2023-01-01 +latex: \scriptsize \sin(a \pm b) = \sin(a) \cos(b) \pm \cos(a) \sin(b) +--- + +For arbitrary angles $\small a$ and $\small b$, the following are true: + +$$ \sin(a + b) = \sin(a) \cos(b) + \cos(a) \sin(b) $$ +$$ \sin(a - b) = \sin(a) \cos(b) - \cos(a) \sin(b) $$ + +$$ \cos(a + b) = \cos(a) \cos(b) - \sin(a) \sin(b) $$ +$$ \cos(a - b) = \cos(a) \cos(b) + \sin(a) \sin(b) $$ + +$$ \tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a) \tan(b)} $$ +$$ \tan(a - b) = \frac{\tan(a) - \tan(b)}{1 + \tan(a) \tan(b)} $$ + +## Sources: +- [Wikipedia](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Angle_sum_and_difference_identities) +- [Byju's](https://byjus.com/maths/trigonometric-identities/) \ No newline at end of file diff --git a/stemformulas-next/src/content/formulas/trig-in-terms-of-e.mdx b/stemformulas-next/src/content/formulas/trig-in-terms-of-e.mdx new file mode 100644 index 00000000..37ebdbc9 --- /dev/null +++ b/stemformulas-next/src/content/formulas/trig-in-terms-of-e.mdx @@ -0,0 +1,18 @@ +--- +title: "Trig functions in terms of e" +description: "Common trig functions in terms of e" +summary: "Common trig functions in terms of e" +tags: ["math", "trigonometry", "e", "complex analysis", "imaginary numbers"] +date: 2022-11-28 +latex: \small{\sin(x) = \frac{e^{ix} - e^{-ix}}{2i}, \cos(x) = \frac{e^{ix} + e^{-ix}}{2}} +--- +$$\small \sin(x) = \frac{e^{ix} - e^{-ix}}{2i}, \cos(x) = \frac{e^{ix} + e^{-ix}}{2}, \tan(x) = \frac{e^{ix} - e^{-ix}}{i(e^{ix} + e^{-ix})}$$ + +Where +* $e$ is [Euler's number](https://en.wikipedia.org/wiki/E_(mathematical_constant)) ~= 2.71828, and +* $i$ is the imaginary unit, $\sqrt{-1}$ + +## Sources +- [Duke.edu](http://webhome.phy.duke.edu/~rgb/Class/phy51/phy51/node15.html) +- [ProofWiki](https://proofwiki.org/wiki/Tangent_Exponential_Formulation) + diff --git a/stemformulas-next/src/content/formulas/vector-calculus-product-rules.mdx b/stemformulas-next/src/content/formulas/vector-calculus-product-rules.mdx new file mode 100644 index 00000000..5a22b78d --- /dev/null +++ b/stemformulas-next/src/content/formulas/vector-calculus-product-rules.mdx @@ -0,0 +1,26 @@ +--- +title: Vector Calculus Product Rules +description: "The product rules in vector calculus." +summary: "The product rules in vector calculus." +tags: ["mathematics", "vector calculus"] +date: 2024-04-19 +latex: "\\nabla (fg) = f(\\nabla g) + g(\\nabla f)" +--- + +For scalar functions $\small f $ and $\small g $ and vector functions $\small \mathbf{A} $ and $\small \mathbf{B} $, the product rules in vector calculus are as follows: + +## Gradient +$$\small \nabla (fg) = f(\nabla g) + g(\nabla f) $$ +$$\small \nabla (\mathbf{A} \cdot \mathbf{B}) = \mathbf{A} \times (\nabla \times \mathbf{B}) + \mathbf{B} \times (\nabla \times \mathbf{A}) + (\mathbf{A} \cdot \nabla) \mathbf{B} + (\mathbf{B} \cdot \nabla) \mathbf{A} $$ + +## Divergence +$$\small \nabla \cdot (f \mathbf{A}) = f(\nabla \cdot \mathbf{A}) + \mathbf{A} \cdot (\nabla f) $$ +$$\small \nabla \cdot (\mathbf{A} \times \mathbf{B}) = \mathbf{B} \cdot (\nabla \times \mathbf{A}) - \mathbf{A} \cdot (\nabla \times \mathbf{B}) $$ + +## Curl +$$\small \nabla \times (f \mathbf{A}) = f(\nabla \times \mathbf{A}) + \mathbf{A} \times (\nabla f) $$ +$$\small \nabla \times (\mathbf{A} \times \mathbf{B}) = (\mathbf{B} \cdot \nabla) \mathbf{A} - (\mathbf{A} \cdot \nabla) \mathbf{B} + \mathbf{A} (\nabla \cdot \mathbf{B}) - \mathbf{B} (\nabla \cdot \mathbf{A}) $$ + +## Sources + +- [Griffith's Introduction to Electrodynamics](https://www.amazon.com/Introduction-Electrodynamics-David-J-Griffiths/dp/1108420419) diff --git a/stemformulas-next/src/content/formulas/viscosity.mdx b/stemformulas-next/src/content/formulas/viscosity.mdx new file mode 100644 index 00000000..c1ad4e74 --- /dev/null +++ b/stemformulas-next/src/content/formulas/viscosity.mdx @@ -0,0 +1,26 @@ +--- +title: Viscosity +description: "The relationship between viscosity and shear deformation rate in a fluid." +summary: "The relationship between viscosity and shear deformation rate in a fluid." +tags: ["physics", "fluid mechanics"] +date: 2024-02-07 +latex: \tau = \mu \frac{\partial u}{\partial y} +--- + +The formula relating viscosity and shear deformation rate in a fluid, also known as Newton's law of viscosity is given by the following equation: + +$$ \tau = \mu \frac{\partial u}{\partial y} $$ + +Where + +* $ \small \tau\ $ is the applied shear stress, +* $ \small \mu $ is the dynamic viscosity, with units of pressure $ \cdot $ time, +* $ \small \frac{\partial u}{\partial y} $ is shear deformation rate, and +* $ \small y $ is the perpendicular distance from the surface. + +In the case where velocity varies linearly with y, the equation simplifies to: + +$$ \tau = \mu \frac{u}{y} $$ + +## Sources +- [Wikipedia](https://en.wikipedia.org/wiki/Viscosity) From f178c90a28a6db2ef676e52c123be89b8e68a838 Mon Sep 17 00:00:00 2001 From: Kevin Lin Date: Mon, 27 Jul 2026 00:24:50 -0700 Subject: [PATCH 03/16] formulas page, styling, --- stemformulas-next/package-lock.json | 109 ++++++++++++++++++ stemformulas-next/package.json | 1 + .../src/app/formulas/[slug]/page.module.css | 16 +++ .../src/app/formulas/[slug]/page.tsx | 14 ++- .../src/app/formulas/page.module.css | 46 ++++++++ stemformulas-next/src/app/formulas/page.tsx | 76 ++++++++++++ stemformulas-next/src/app/globals.css | 94 ++++++++++++--- .../src/content/formulas/bayes-theorem.mdx | 2 +- .../src/content/formulas/curl.mdx | 3 + .../src/content/formulas/determinant.mdx | 1 + .../formulas/eulers-identity-formula.mdx | 1 + .../src/content/formulas/feynmanns-trick.mdx | 1 + .../fundmantal-theorem-of-calculus.mdx | 1 + .../formulas/incompressible-navier-stokes.mdx | 6 + .../content/formulas/laplaces-equation.mdx | 2 + .../src/content/formulas/log-laws.mdx | 5 + .../formulas/lorentz-transformations.mdx | 3 + .../content/formulas/maxwells-equations.mdx | 9 ++ .../content/formulas/newtonian-momentum.mdx | 1 + .../src/content/formulas/newtons-2nd-law.mdx | 1 + .../content/formulas/pythagorean-theorem.mdx | 1 + .../content/formulas/quadratic-formula.mdx | 1 + .../src/content/formulas/quadric-surfaces.mdx | 1 + .../src/content/formulas/specific-gravity.mdx | 1 + .../formulas/taylor-expansion-of-e.mdx | 1 + .../trig-angle-sum-difference-identities.mdx | 3 + .../content/formulas/trig-in-terms-of-e.mdx | 1 + .../vector-calculus-product-rules.mdx | 6 + 28 files changed, 381 insertions(+), 26 deletions(-) create mode 100644 stemformulas-next/src/app/formulas/page.module.css create mode 100644 stemformulas-next/src/app/formulas/page.tsx diff --git a/stemformulas-next/package-lock.json b/stemformulas-next/package-lock.json index 5b250ea3..96fb36d6 100644 --- a/stemformulas-next/package-lock.json +++ b/stemformulas-next/package-lock.json @@ -8,6 +8,7 @@ "name": "stemformulas-next", "version": "0.1.0", "dependencies": { + "gray-matter": "^4.0.3", "next": "16.2.12", "next-mdx-remote": "^6.0.0", "react": "19.2.4", @@ -3588,6 +3589,19 @@ "url": "https://opencollective.com/eslint" } }, + "node_modules/esprima": { + "version": "4.0.1", + "resolved": "https://registry.npmjs.org/esprima/-/esprima-4.0.1.tgz", + "integrity": "sha512-eGuFFw7Upda+g4p+QHvnW0RyTX/SVeJBDM/gCtMARO0cLuT2HcEKnTPvhjV6aGeqrCB/sbNop0Kszm0jsaWU4A==", + "license": "BSD-2-Clause", + "bin": { + "esparse": "bin/esparse.js", + "esvalidate": "bin/esvalidate.js" + }, + "engines": { + "node": ">=4" + } + }, "node_modules/esquery": { "version": "1.7.0", "resolved": "https://registry.npmjs.org/esquery/-/esquery-1.7.0.tgz", @@ -3731,6 +3745,18 @@ "integrity": "sha512-fjquC59cD7CyW6urNXK0FBufkZcoiGG80wTuPujX590cB5Ttln20E2UB4S/WARVqhXffZl2LNgS+gQdPIIim/g==", "license": "MIT" }, + "node_modules/extend-shallow": { + "version": "2.0.1", + "resolved": "https://registry.npmjs.org/extend-shallow/-/extend-shallow-2.0.1.tgz", + "integrity": "sha512-zCnTtlxNoAiDc3gqY2aYAWFx7XWWiasuF2K8Me5WbN8otHKTUKBwjPtNpRs/rbUZm7KxWAaNj7P1a/p52GbVug==", + "license": "MIT", + "dependencies": { + "is-extendable": "^0.1.0" + }, + "engines": { + "node": ">=0.10.0" + } + }, "node_modules/fast-deep-equal": { "version": "3.1.3", "resolved": "https://registry.npmjs.org/fast-deep-equal/-/fast-deep-equal-3.1.3.tgz", @@ -4062,6 +4088,43 @@ "url": "https://github.com/sponsors/ljharb" } }, + "node_modules/gray-matter": { + "version": "4.0.3", + "resolved": "https://registry.npmjs.org/gray-matter/-/gray-matter-4.0.3.tgz", + "integrity": "sha512-5v6yZd4JK3eMI3FqqCouswVqwugaA9r4dNZB1wwcmrD02QkV5H0y7XBQW8QwQqEaZY1pM9aqORSORhJRdNK44Q==", + "license": "MIT", + "dependencies": { + "js-yaml": "^3.13.1", + "kind-of": "^6.0.2", + "section-matter": "^1.0.0", + "strip-bom-string": "^1.0.0" + }, + "engines": { + "node": ">=6.0" + } + }, + "node_modules/gray-matter/node_modules/argparse": { + "version": "1.0.10", + "resolved": "https://registry.npmjs.org/argparse/-/argparse-1.0.10.tgz", + "integrity": "sha512-o5Roy6tNG4SL/FOkCAN6RzjiakZS25RLYFrcMttJqbdd8BWrnA+fGz57iN5Pb06pvBGvl5gQ0B48dJlslXvoTg==", + "license": "MIT", + "dependencies": { + "sprintf-js": "~1.0.2" + } + }, + "node_modules/gray-matter/node_modules/js-yaml": { + "version": "3.15.0", + "resolved": "https://registry.npmjs.org/js-yaml/-/js-yaml-3.15.0.tgz", + "integrity": "sha512-ttBQIIQPDeLjpPOohtUdXuXUVoA2uIB6fEH9HyJ7234s5mBJ5wTx20njxplLZQgLaOfpmPQA7X2t5AX6tIPbog==", + "license": "MIT", + "dependencies": { + "argparse": "^1.0.7", + "esprima": "^4.0.0" + }, + "bin": { + "js-yaml": "bin/js-yaml.js" + } + }, "node_modules/has-bigints": { "version": "1.1.0", "resolved": "https://registry.npmjs.org/has-bigints/-/has-bigints-1.1.0.tgz", @@ -4635,6 +4698,15 @@ "url": "https://github.com/sponsors/ljharb" } }, + "node_modules/is-extendable": { + "version": "0.1.1", + "resolved": "https://registry.npmjs.org/is-extendable/-/is-extendable-0.1.1.tgz", + "integrity": "sha512-5BMULNob1vgFX6EjQw5izWDxrecWK9AM72rugNr0TFldMOi0fj6Jk+zeKIt0xGj4cEfQIJth4w3OKWOJ4f+AFw==", + "license": "MIT", + "engines": { + "node": ">=0.10.0" + } + }, "node_modules/is-extglob": { "version": "2.1.1", "resolved": "https://registry.npmjs.org/is-extglob/-/is-extglob-2.1.1.tgz", @@ -5064,6 +5136,15 @@ "json-buffer": "3.0.1" } }, + "node_modules/kind-of": { + "version": "6.0.3", + "resolved": "https://registry.npmjs.org/kind-of/-/kind-of-6.0.3.tgz", + "integrity": "sha512-dcS1ul+9tmeD95T+x28/ehLgd9mENa3LsvDTtzm3vyBEO7RPptvAD+t44WVXaUjTBRcrpFeFlC8WCruUR456hw==", + "license": "MIT", + "engines": { + "node": ">=0.10.0" + } + }, "node_modules/language-subtag-registry": { "version": "0.3.23", "resolved": "https://registry.npmjs.org/language-subtag-registry/-/language-subtag-registry-0.3.23.tgz", @@ -6943,6 +7024,19 @@ "integrity": "sha512-eNv+WrVbKu1f3vbYJT/xtiF5syA5HPIMtf9IgY/nKg0sWqzAUEvqY/xm7OcZc/qafLx/iO9FgOmeSAp4v5ti/Q==", "license": "MIT" }, + "node_modules/section-matter": { + "version": "1.0.0", + "resolved": "https://registry.npmjs.org/section-matter/-/section-matter-1.0.0.tgz", + "integrity": "sha512-vfD3pmTzGpufjScBh50YHKzEu2lxBWhVEHsNGoEXmCmn2hKGfeNLYMzCJpe8cD7gqX7TJluOVpBkAequ6dgMmA==", + "license": "MIT", + "dependencies": { + "extend-shallow": "^2.0.1", + "kind-of": "^6.0.0" + }, + "engines": { + "node": ">=4" + } + }, "node_modules/semver": { "version": "6.3.1", "resolved": "https://registry.npmjs.org/semver/-/semver-6.3.1.tgz", @@ -7187,6 +7281,12 @@ "url": "https://github.com/sponsors/wooorm" } }, + "node_modules/sprintf-js": { + "version": "1.0.3", + "resolved": "https://registry.npmjs.org/sprintf-js/-/sprintf-js-1.0.3.tgz", + "integrity": "sha512-D9cPgkvLlV3t3IzL0D0YLvGA9Ahk4PcvVwUbN0dSGr1aP0Nrt4AEnTUbuGvquEC0mA64Gqt1fzirlRs5ibXx8g==", + "license": "BSD-3-Clause" + }, "node_modules/stable-hash": { "version": "0.0.5", "resolved": "https://registry.npmjs.org/stable-hash/-/stable-hash-0.0.5.tgz", @@ -7346,6 +7446,15 @@ "node": ">=4" } }, + "node_modules/strip-bom-string": { + "version": "1.0.0", + "resolved": "https://registry.npmjs.org/strip-bom-string/-/strip-bom-string-1.0.0.tgz", + "integrity": "sha512-uCC2VHvQRYu+lMh4My/sFNmF2klFymLX1wHJeXnbEJERpV/ZsVuonzerjfrGpIGF7LBVa1O7i9kjiWvJiFck8g==", + "license": "MIT", + "engines": { + "node": ">=0.10.0" + } + }, "node_modules/strip-json-comments": { "version": "3.1.1", "resolved": "https://registry.npmjs.org/strip-json-comments/-/strip-json-comments-3.1.1.tgz", diff --git a/stemformulas-next/package.json b/stemformulas-next/package.json index 016b3aad..2d9d3008 100644 --- a/stemformulas-next/package.json +++ b/stemformulas-next/package.json @@ -9,6 +9,7 @@ "lint": "eslint" }, "dependencies": { + "gray-matter": "^4.0.3", "next": "16.2.12", "next-mdx-remote": "^6.0.0", "react": "19.2.4", diff --git a/stemformulas-next/src/app/formulas/[slug]/page.module.css b/stemformulas-next/src/app/formulas/[slug]/page.module.css index b30f6203..9b97b7c7 100644 --- a/stemformulas-next/src/app/formulas/[slug]/page.module.css +++ b/stemformulas-next/src/app/formulas/[slug]/page.module.css @@ -3,6 +3,15 @@ margin-bottom: 0; } +.title { + font-size: 2.25rem; + font-weight: 800; + line-height: 2.5rem; + color: var(--color-neutral-900); + margin: 0; + padding: 0; +} + .articleLayout { display: flex; flex-direction: column; @@ -73,6 +82,13 @@ } .tags { + margin: 1rem 0; + font-size: 1rem; + line-height: 1.5rem; + color: var(--color-neutral-500); +} + +.tagsInner { margin: 0.5rem 0; font-size: 0.75rem; color: var(--color-neutral-500); diff --git a/stemformulas-next/src/app/formulas/[slug]/page.tsx b/stemformulas-next/src/app/formulas/[slug]/page.tsx index 35f0f343..a9efee3d 100644 --- a/stemformulas-next/src/app/formulas/[slug]/page.tsx +++ b/stemformulas-next/src/app/formulas/[slug]/page.tsx @@ -77,14 +77,16 @@ export default async function FormulaPage({ return (

-

{formula.frontmatter.title}

+

{formula.frontmatter.title}

{formula.frontmatter.tags && formula.frontmatter.tags.length > 0 && (
- {formula.frontmatter.tags.map((tag) => ( - - {tag} - - ))} +
+ {formula.frontmatter.tags.map((tag) => ( + + {tag} + + ))} +
)}
diff --git a/stemformulas-next/src/app/formulas/page.module.css b/stemformulas-next/src/app/formulas/page.module.css new file mode 100644 index 00000000..ce77c8b2 --- /dev/null +++ b/stemformulas-next/src/app/formulas/page.module.css @@ -0,0 +1,46 @@ +.heading { + font-size: 2.25rem; + font-weight: 800; + margin-top: 0; +} + +.subtitle { + margin-top: 0.5rem; + font-size: 0.875rem; + color: var(--color-neutral-900); +} + +.grid { + margin-top: 1.5rem; +} + +.card { + display: flex; + flex-direction: column; + align-items: center; + justify-content: center; + gap: 0.25rem; + border: 1px solid darkgray; + border-radius: 12px; + padding: 0.75rem; + overflow: hidden; + text-decoration: none; + color: inherit; + max-width: 100%; +} + +.card:hover { + background: var(--color-primary-100); +} + +.cardTitle { + font-size: 1.125rem; + font-weight: 600; + text-align: center; + color: var(--color-neutral-800); +} + +.cardLatex { + font-size: 0.8125rem; + color: var(--color-neutral-500); +} diff --git a/stemformulas-next/src/app/formulas/page.tsx b/stemformulas-next/src/app/formulas/page.tsx new file mode 100644 index 00000000..7bf40812 --- /dev/null +++ b/stemformulas-next/src/app/formulas/page.tsx @@ -0,0 +1,76 @@ +import fs from "fs"; +import path from "path"; +import Link from "next/link"; +import katex from "katex"; +import matter from "gray-matter"; +import styles from "./page.module.css"; + +interface FormulaMeta { + slug: string; + title: string; + latex: string; +} + +function getFormulas(): FormulaMeta[] { + const dir = path.join(process.cwd(), "src/content/formulas"); + const files = fs.readdirSync(dir).filter((f) => f.endsWith(".mdx")); + + return files.map((file) => { + const slug = file.replace(".mdx", ""); + const source = fs.readFileSync(path.join(dir, file), "utf-8"); + const { data } = matter(source); + return { + slug, + title: data.title || slug, + latex: data.latex || "", + }; + }); +} + +export default function FormulasPage() { + const formulas = getFormulas(); + + return ( + <> +
+

formulas

+
+ +

+ Click on any formula to visit its page for more details. +

+ +
+ {formulas.map((formula) => { + let renderedLatex = ""; + if (formula.latex) { + try { + renderedLatex = katex.renderToString(formula.latex, { + throwOnError: false, + displayMode: true, + }); + } catch { + renderedLatex = formula.latex; + } + } + + return ( + +

{formula.title}

+ {renderedLatex && ( +
+ )} + + ); + })} +
+ + ); +} diff --git a/stemformulas-next/src/app/globals.css b/stemformulas-next/src/app/globals.css index 460ea211..77ca32a4 100644 --- a/stemformulas-next/src/app/globals.css +++ b/stemformulas-next/src/app/globals.css @@ -9,6 +9,7 @@ --color-neutral-700: #374151; --color-neutral-800: #1f2937; --color-neutral-900: #111827; + --color-primary-100: #f1f5f9; --color-primary-500: #64748b; --color-primary-600: #475569; --color-primary-700: #334155; @@ -27,7 +28,7 @@ body { padding: 0 1.5rem; font-family: ui-sans-serif, system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, "Helvetica Neue", Arial, "Noto Sans", sans-serif; - font-size: 1.125rem; + font-size: 1rem; line-height: 1.75rem; color: var(--color-neutral-900); background: #fff; @@ -35,10 +36,34 @@ body { -webkit-font-smoothing: antialiased; } +@media (min-width: 640px) { + body { + padding: 0 3.5rem; + } +} + +@media (min-width: 768px) { + body { + padding: 0 6rem; + } +} + +@media (min-width: 1024px) { + body { + padding: 0 8rem; + } +} + * { box-sizing: border-box; } +article, +header, +section { + margin: 0; +} + a { color: inherit; text-decoration: none; @@ -165,28 +190,40 @@ a { max-width: 65ch; } -.prose h1 { - font-size: 2.25rem; - font-weight: 800; - color: var(--color-neutral-900); +.prose h2 { + font-size: 1.5em; + font-weight: 700; + line-height: 1.3333; + margin-top: 2em; + margin-bottom: 1em; +} + +.prose h3 { + font-size: 1.25em; + font-weight: 600; + line-height: 1.6; + margin-top: 1.6em; + margin-bottom: 0.6em; +} + +.prose > :first-child { margin-top: 0; } -.prose h2 { - font-size: 1.5rem; - font-weight: 700; - margin-top: 2rem; - margin-bottom: 0.75rem; +.prose > :last-child { + margin-bottom: 0; } .prose p { - margin-bottom: 1rem; + margin-top: 1.25em; + margin-bottom: 1.25em; } .prose ul, .prose ol { - padding-left: 1.5rem; - margin-bottom: 1rem; + padding-left: 1.625em; + margin-top: 1.25em; + margin-bottom: 1.25em; } .prose ul { @@ -198,7 +235,12 @@ a { } .prose li { - margin-bottom: 0.25rem; + margin-top: 0.5em; + margin-bottom: 0.5em; +} + +.prose li > p { + margin: 0; } .prose a { @@ -222,6 +264,11 @@ a { width: 400px; } +.prose iframe { + max-width: 100%; + border-radius: 0.375rem; +} + .prose blockquote { border-left: 4px solid var(--color-neutral-300); padding-left: 1rem; @@ -332,14 +379,23 @@ a { } /* ── KaTeX ── */ -.katex-html { +/* Display math:

containing only a .katex element */ +p:has(> .katex:first-child:last-child) { + display: block; + margin: 1em 0; + text-align: center; font-size: 1.2em; - color: #000; } -.katex-display > .katex { - padding-left: 0.5em !important; +p:has(> .katex:first-child:last-child) .katex { + display: inline-block; text-align: left; + font-size: 1.21em; +} + +.katex-html { + font-size: 1.2em; + color: #000; } .katex { @@ -350,7 +406,7 @@ a { .grid-container { display: grid; grid-template-columns: 1fr; - gap: 15px; + gap: 12px; } @media (min-width: 640px) { diff --git a/stemformulas-next/src/content/formulas/bayes-theorem.mdx b/stemformulas-next/src/content/formulas/bayes-theorem.mdx index c66cc754..f68e6866 100644 --- a/stemformulas-next/src/content/formulas/bayes-theorem.mdx +++ b/stemformulas-next/src/content/formulas/bayes-theorem.mdx @@ -30,7 +30,7 @@ Where $\small \neg A$ is the complement of $\small A$. This video by 3Blue1Brown was the single biggest help in my understanding of Bayes' Theorem, you should definitely watch it if you're struggling to understand it: -{{< youtube id="HZGCoVF3YvM" >}} + ## Sources diff --git a/stemformulas-next/src/content/formulas/curl.mdx b/stemformulas-next/src/content/formulas/curl.mdx index d189c2ca..bc032fa2 100644 --- a/stemformulas-next/src/content/formulas/curl.mdx +++ b/stemformulas-next/src/content/formulas/curl.mdx @@ -15,6 +15,7 @@ In cartesian coordinates, the curl of a vector field $ \small \mathbf{F} $ is gi $$ \nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\\\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\\\ F_x & F_y & F_z \end{vmatrix} + $$ Where: @@ -35,6 +36,7 @@ In cylindrical coordinates, the curl of a vector field $ \small \mathbf{F} $ is $$ \nabla \times \mathbf{F} = \left( \frac{1}{\rho} \frac{\partial F_z}{\partial \varphi} - \frac{\partial F_\varphi}{\partial z} \right) \hat{\rho} + \left( \frac{\partial F_\rho}{\partial z} - \frac{\partial F_z}{\partial \rho} \right) \hat{\varphi} + \frac{1}{\rho} \left( \frac{\partial}{\partial \rho} \left( \rho F_\varphi \right) - \frac{\partial F_\rho}{\partial \varphi} \right) \hat{z} + $$ Where: @@ -48,6 +50,7 @@ In spherical coordinates, the curl of a vector field $ \small \mathbf{F} $ is gi $$ \nabla \times \mathbf{F} = \frac{1}{r \sin \theta} \left( \frac{\partial (\sin \theta F\_\varphi)}{\partial \theta} - \frac{\partial F\_\theta}{\partial \varphi} \right) \hat{r} + \frac{1}{r} \left( \frac{1}{\sin \theta} \frac{\partial F\_r}{\partial \varphi} - \frac{\partial (r F_\varphi)}{\partial r} \right) \hat{\theta} + \frac{1}{r} \left( \frac{\partial (r F\_\theta)}{\partial r} - \frac{\partial F\_r}{\partial \theta} \right) \hat{\varphi} + $$ Where: diff --git a/stemformulas-next/src/content/formulas/determinant.mdx b/stemformulas-next/src/content/formulas/determinant.mdx index eea2f2b4..c29f8692 100644 --- a/stemformulas-next/src/content/formulas/determinant.mdx +++ b/stemformulas-next/src/content/formulas/determinant.mdx @@ -14,6 +14,7 @@ $$ \det\begin{bmatrix}\\ c & d\\\\ \end{bmatrix}=ad-cb $$ + $$\det\begin{bmatrix} a & b & c\\\\ d & e & f\\\\ diff --git a/stemformulas-next/src/content/formulas/eulers-identity-formula.mdx b/stemformulas-next/src/content/formulas/eulers-identity-formula.mdx index 86bb28cc..22a04dfb 100644 --- a/stemformulas-next/src/content/formulas/eulers-identity-formula.mdx +++ b/stemformulas-next/src/content/formulas/eulers-identity-formula.mdx @@ -7,6 +7,7 @@ date: 2022-11-28 latex: e^{i\pi} = -1 --- ## Euler's Identity + $$e^{i\pi} = -1$$ Where diff --git a/stemformulas-next/src/content/formulas/feynmanns-trick.mdx b/stemformulas-next/src/content/formulas/feynmanns-trick.mdx index 415e649f..39e09954 100644 --- a/stemformulas-next/src/content/formulas/feynmanns-trick.mdx +++ b/stemformulas-next/src/content/formulas/feynmanns-trick.mdx @@ -7,6 +7,7 @@ date: 2023-03-16 latex: \int_0^{\infty} x^n e^{-tx} dx = \frac{n!}{t^{n+1}} --- ## Feynmann's Trick For Exponential Integrals + $$\int_0^{\infty} x^n e^{-tx} dx = \frac{n!}{t^{n+1}}$$ Where diff --git a/stemformulas-next/src/content/formulas/fundmantal-theorem-of-calculus.mdx b/stemformulas-next/src/content/formulas/fundmantal-theorem-of-calculus.mdx index 6f3aa55d..3a1f6a86 100644 --- a/stemformulas-next/src/content/formulas/fundmantal-theorem-of-calculus.mdx +++ b/stemformulas-next/src/content/formulas/fundmantal-theorem-of-calculus.mdx @@ -6,6 +6,7 @@ tags: ["math", "calculus"] date: 2022-12-23 latex: \int_a^b f(x) dx = F(b) - F(a) --- + $$\int_a^b f(x) dx = F(b) - F(a)$$ Where diff --git a/stemformulas-next/src/content/formulas/incompressible-navier-stokes.mdx b/stemformulas-next/src/content/formulas/incompressible-navier-stokes.mdx index c5f1af67..8de1064e 100644 --- a/stemformulas-next/src/content/formulas/incompressible-navier-stokes.mdx +++ b/stemformulas-next/src/content/formulas/incompressible-navier-stokes.mdx @@ -14,15 +14,19 @@ The incompressible Navier-Stokes equations are a simpler form of the [Navier-St For cartesian coordinates $ \small x, y, z $ and time $ \small t $, the incompressible Navier-Stokes equations are: ### Continuity Equation + $$ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 $$ ### X-momentum Equation + $$ \rho \left(\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z}\right) = - \frac{\partial P}{\partial x} + \mu \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) + \rho g_x$$ ### Y-momentum Equation + $$ \rho \left(\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z}\right) = - \frac{\partial P}{\partial y} + \mu \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 v}{\partial z^2} \right) + \rho g_y$$ ### Z-momentum Equation + $$ \rho \left(\frac{\partial w}{\partial t} + u \frac{\partial w}{\partial x} + v \frac{\partial w}{\partial y} + w \frac{\partial w}{\partial z}\right) = - \frac{\partial P}{\partial z} + \mu \left( \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2} \right) + \rho g_z$$ Where @@ -37,9 +41,11 @@ Where For cylindrical coordinates $ \small r, \theta, z $ and time $ \small t $, the incompressible Navier-Stokes equations are: ### Continuity Equation + $$ \frac{1}{r} \frac{\partial (r u_r)}{\partial r} + \frac{1}{r} \frac{\partial (u_{\theta})}{\partial \theta} + \frac{\partial (u_{z})}{\partial z} = 0 $$ ### R-component + $$ \rho \left(\frac{\partial u_r}{\partial t} + u_r \frac{\partial u_r}{\partial r} + \frac{u_{\theta}}{r} \frac{\partial u_r}{\partial \theta} - \frac{u_{\theta}^2}{r} + u_z \frac{\partial u_r}{\partial z} \right) = - \frac{\partial P}{\partial r} + \rho g_r+ \mu \left[ \frac{1}{r} \frac{\partial}{\partial r} \left(r \frac{\partial u_r}{\partial r}\right) - \frac{u_r}{r^2} + \frac{1}{r^2} \frac{\partial^2 u_r}{\partial \theta^2} -\frac{2}{r^2} \frac{\partial u_{\theta}}{\partial \theta} + \frac{\partial^2 u_r}{\partial z^2} \right] $$ ### $\small \theta$-component diff --git a/stemformulas-next/src/content/formulas/laplaces-equation.mdx b/stemformulas-next/src/content/formulas/laplaces-equation.mdx index f1888afe..6b70b519 100644 --- a/stemformulas-next/src/content/formulas/laplaces-equation.mdx +++ b/stemformulas-next/src/content/formulas/laplaces-equation.mdx @@ -22,9 +22,11 @@ Where $$ \scriptsize \nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} = 0$$ #### In cylindrical coordinates: + $$ \scriptsize \nabla^2 f = \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial f}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 f}{\partial \phi^2} + \frac{\partial^2 f}{\partial z^2} = 0$$ #### In spherical coordinates: + $$ \scriptsize \nabla^2 f = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial f}{\partial r} \right) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial f}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta}\frac{\partial^2 f}{\partial \varphi^2} = 0$$ ## Sources diff --git a/stemformulas-next/src/content/formulas/log-laws.mdx b/stemformulas-next/src/content/formulas/log-laws.mdx index 177b8eaf..dd3b5eaa 100644 --- a/stemformulas-next/src/content/formulas/log-laws.mdx +++ b/stemformulas-next/src/content/formulas/log-laws.mdx @@ -13,10 +13,15 @@ Logarithms are the inverse of exponentials. They are useful for solving equation For any positive real number $ \small b \neq 1 $, and positive real numbers $ \small a $, and $ \small c $, the following properties hold: $$ \log_b(1) = 0 $$ + $$ \log_b(b) = 1 $$ + $$ \log_b(a \cdot c) = \log_b(a) + \log_b(c) $$ + $$ \log_b \left(\frac{a}{c} \right) = \log_b(a) - \log_b(c) $$ + $$ \log_b(a^c) = c \cdot \log_b(a) $$ + $$ b^{\log_b(k)} = k $$ Note that $ k $ can be any real number in the last equation. diff --git a/stemformulas-next/src/content/formulas/lorentz-transformations.mdx b/stemformulas-next/src/content/formulas/lorentz-transformations.mdx index 363953f5..4287cd1b 100644 --- a/stemformulas-next/src/content/formulas/lorentz-transformations.mdx +++ b/stemformulas-next/src/content/formulas/lorentz-transformations.mdx @@ -10,8 +10,11 @@ latex: x' = \frac{x-vt}{\sqrt{1-v^2/c^2}} For a fixed frame of reference $\small x, y, z, t$, the following transformations can be used to transform to a frame of reference moving with velocity $\small v$ with respect to the fixed frame: $$\small x' = \frac{x-vt}{\sqrt{1-v^2/c^2}}$$ + $$ \small y' = y$$ + $$ \small z' = z$$ + $$ \small t' = \frac{t-vx/c^2}{\sqrt{1-v^2/c^2}}$$ Where diff --git a/stemformulas-next/src/content/formulas/maxwells-equations.mdx b/stemformulas-next/src/content/formulas/maxwells-equations.mdx index 0eaeb4fe..3d456319 100644 --- a/stemformulas-next/src/content/formulas/maxwells-equations.mdx +++ b/stemformulas-next/src/content/formulas/maxwells-equations.mdx @@ -36,8 +36,11 @@ Respectively, these equations are Gauss's Law, Gauss's Law for Magnetism, Farada In a vacuum (where there is no charge or current), these simplify to: $$ \nabla \cdot \mathbf{E} = 0 $$ + $$ \nabla \cdot \mathbf{B} = 0 $$ + $$ \nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}$$ + $$ \nabla \times \mathbf{B} = \mu\_0 \varepsilon\_0 \frac{\partial \mathbf{E}}{\partial t} $$ @@ -46,8 +49,11 @@ $$ \nabla \times \mathbf{B} = \mu\_0 \varepsilon\_0 \frac{\partial \mathbf{E}}{\ In integral form, Maxwell's equations are given by: $$ \oiint\_{\partial \Omega} \mathbf{E} \cdot d\mathbf{S} = \frac{Q}{\varepsilon\_0} = \frac{1}{\varepsilon\_0}\iiint_{\Omega} \rho \ dV $$ + $$ \oiint\_{\partial \Omega} \mathbf{B} \cdot d\mathbf{S} = 0 $$ + $$ \oint\_{\partial \Sigma} \mathbf{E} \cdot d\mathbf{l} = - \frac{\text{d}}{\text{d} t} \iint\_{\Sigma} \mathbf{B} \cdot d\mathbf{S} $$ + $$ \oint\_{\partial \Sigma} \mathbf{B} \cdot d\mathbf{l} = \mu\_0 \iint\_{\Sigma} \mathbf{J} \cdot d\mathbf{S} + \mu\_0 \varepsilon\_0 \frac{\text{d}}{\text{d} t} \iint\_{\Sigma} \mathbf{E} \cdot d\mathbf{S} $$ Where: @@ -68,8 +74,11 @@ Where: In a vacuum (where there is no charge or current), these simplify to: $$ \oiint\_{\partial \Omega} \mathbf{E} \cdot d\mathbf{S} = 0 $$ + $$ \oiint\_{\partial \Omega} \mathbf{B} \cdot d\mathbf{S} = 0 $$ + $$ \oint\_{\partial \Sigma} \mathbf{E} \cdot d\mathbf{l} = - \frac{\text{d}}{\text{d} t} \iint\_{\Sigma} \mathbf{B} \cdot d\mathbf{S} $$ + $$ \oint\_{\partial \Sigma} \mathbf{B} \cdot d\mathbf{l} = \mu\_0 \varepsilon\_0 \frac{\text{d}}{\text{d} t} \iint\_{\Sigma} \mathbf{E} \cdot d\mathbf{S} $$ ## Sources diff --git a/stemformulas-next/src/content/formulas/newtonian-momentum.mdx b/stemformulas-next/src/content/formulas/newtonian-momentum.mdx index 13281c6d..dbd92f24 100644 --- a/stemformulas-next/src/content/formulas/newtonian-momentum.mdx +++ b/stemformulas-next/src/content/formulas/newtonian-momentum.mdx @@ -6,6 +6,7 @@ tags: ["physics", "mass", "velocity"] date: 2022-12-08 latex: p = mv --- + $$p = mv$$ Where diff --git a/stemformulas-next/src/content/formulas/newtons-2nd-law.mdx b/stemformulas-next/src/content/formulas/newtons-2nd-law.mdx index 4e4a06a6..45432d80 100644 --- a/stemformulas-next/src/content/formulas/newtons-2nd-law.mdx +++ b/stemformulas-next/src/content/formulas/newtons-2nd-law.mdx @@ -6,6 +6,7 @@ tags: ["physics", "mechanics"] date: 2022-12-04 latex: F = ma --- + $$F = ma$$ Where diff --git a/stemformulas-next/src/content/formulas/pythagorean-theorem.mdx b/stemformulas-next/src/content/formulas/pythagorean-theorem.mdx index c751d469..6995aaa2 100644 --- a/stemformulas-next/src/content/formulas/pythagorean-theorem.mdx +++ b/stemformulas-next/src/content/formulas/pythagorean-theorem.mdx @@ -9,6 +9,7 @@ latex: a^2 + b^2 = c^2 For any right triangle *abc* (see diagram below): The lengths of the sides are related by the Pythagorean Theorem: + $$a^2 + b^2 = c^2$$ ![diagram](triangle.png) diff --git a/stemformulas-next/src/content/formulas/quadratic-formula.mdx b/stemformulas-next/src/content/formulas/quadratic-formula.mdx index 679e8ae5..a040b4e2 100644 --- a/stemformulas-next/src/content/formulas/quadratic-formula.mdx +++ b/stemformulas-next/src/content/formulas/quadratic-formula.mdx @@ -6,6 +6,7 @@ tags: ["math", "algebra"] date: 2022-12-06 latex: x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} --- + $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Where diff --git a/stemformulas-next/src/content/formulas/quadric-surfaces.mdx b/stemformulas-next/src/content/formulas/quadric-surfaces.mdx index 28eeff9a..d67ecb8b 100644 --- a/stemformulas-next/src/content/formulas/quadric-surfaces.mdx +++ b/stemformulas-next/src/content/formulas/quadric-surfaces.mdx @@ -29,6 +29,7 @@ $$ {x^2 \over a^2} - {y^2 \over b^2} = z $$ The equation of a hyperboloid of one sheet is: + $$ {x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = 1 $$ ## Hyperboloid of two sheets: diff --git a/stemformulas-next/src/content/formulas/specific-gravity.mdx b/stemformulas-next/src/content/formulas/specific-gravity.mdx index df345911..d8f52ecd 100644 --- a/stemformulas-next/src/content/formulas/specific-gravity.mdx +++ b/stemformulas-next/src/content/formulas/specific-gravity.mdx @@ -8,6 +8,7 @@ latex: SG = \frac{\rho}{\rho_{H_2O}} --- The specific gravity of a substance is the dimensionless ratio of its density to the density of water: + $$ SG = \frac{\rho}{\rho_{H_2O}} $$ Where diff --git a/stemformulas-next/src/content/formulas/taylor-expansion-of-e.mdx b/stemformulas-next/src/content/formulas/taylor-expansion-of-e.mdx index 65ee1d49..c067559c 100644 --- a/stemformulas-next/src/content/formulas/taylor-expansion-of-e.mdx +++ b/stemformulas-next/src/content/formulas/taylor-expansion-of-e.mdx @@ -6,6 +6,7 @@ tags: ["math", "calculus"] date: 2022-12-08 latex: e^x = \sum_{n=0}^\infty \frac{x^n}{n!} --- + $$e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$$ Where diff --git a/stemformulas-next/src/content/formulas/trig-angle-sum-difference-identities.mdx b/stemformulas-next/src/content/formulas/trig-angle-sum-difference-identities.mdx index bd3dfd0b..2d5bfa40 100644 --- a/stemformulas-next/src/content/formulas/trig-angle-sum-difference-identities.mdx +++ b/stemformulas-next/src/content/formulas/trig-angle-sum-difference-identities.mdx @@ -10,12 +10,15 @@ latex: \scriptsize \sin(a \pm b) = \sin(a) \cos(b) \pm \cos(a) \sin(b) For arbitrary angles $\small a$ and $\small b$, the following are true: $$ \sin(a + b) = \sin(a) \cos(b) + \cos(a) \sin(b) $$ + $$ \sin(a - b) = \sin(a) \cos(b) - \cos(a) \sin(b) $$ $$ \cos(a + b) = \cos(a) \cos(b) - \sin(a) \sin(b) $$ + $$ \cos(a - b) = \cos(a) \cos(b) + \sin(a) \sin(b) $$ $$ \tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a) \tan(b)} $$ + $$ \tan(a - b) = \frac{\tan(a) - \tan(b)}{1 + \tan(a) \tan(b)} $$ ## Sources: diff --git a/stemformulas-next/src/content/formulas/trig-in-terms-of-e.mdx b/stemformulas-next/src/content/formulas/trig-in-terms-of-e.mdx index 37ebdbc9..b64ca78e 100644 --- a/stemformulas-next/src/content/formulas/trig-in-terms-of-e.mdx +++ b/stemformulas-next/src/content/formulas/trig-in-terms-of-e.mdx @@ -6,6 +6,7 @@ tags: ["math", "trigonometry", "e", "complex analysis", "imaginary numbers"] date: 2022-11-28 latex: \small{\sin(x) = \frac{e^{ix} - e^{-ix}}{2i}, \cos(x) = \frac{e^{ix} + e^{-ix}}{2}} --- + $$\small \sin(x) = \frac{e^{ix} - e^{-ix}}{2i}, \cos(x) = \frac{e^{ix} + e^{-ix}}{2}, \tan(x) = \frac{e^{ix} - e^{-ix}}{i(e^{ix} + e^{-ix})}$$ Where diff --git a/stemformulas-next/src/content/formulas/vector-calculus-product-rules.mdx b/stemformulas-next/src/content/formulas/vector-calculus-product-rules.mdx index 5a22b78d..f623daf1 100644 --- a/stemformulas-next/src/content/formulas/vector-calculus-product-rules.mdx +++ b/stemformulas-next/src/content/formulas/vector-calculus-product-rules.mdx @@ -10,15 +10,21 @@ latex: "\\nabla (fg) = f(\\nabla g) + g(\\nabla f)" For scalar functions $\small f $ and $\small g $ and vector functions $\small \mathbf{A} $ and $\small \mathbf{B} $, the product rules in vector calculus are as follows: ## Gradient + $$\small \nabla (fg) = f(\nabla g) + g(\nabla f) $$ + $$\small \nabla (\mathbf{A} \cdot \mathbf{B}) = \mathbf{A} \times (\nabla \times \mathbf{B}) + \mathbf{B} \times (\nabla \times \mathbf{A}) + (\mathbf{A} \cdot \nabla) \mathbf{B} + (\mathbf{B} \cdot \nabla) \mathbf{A} $$ ## Divergence + $$\small \nabla \cdot (f \mathbf{A}) = f(\nabla \cdot \mathbf{A}) + \mathbf{A} \cdot (\nabla f) $$ + $$\small \nabla \cdot (\mathbf{A} \times \mathbf{B}) = \mathbf{B} \cdot (\nabla \times \mathbf{A}) - \mathbf{A} \cdot (\nabla \times \mathbf{B}) $$ ## Curl + $$\small \nabla \times (f \mathbf{A}) = f(\nabla \times \mathbf{A}) + \mathbf{A} \times (\nabla f) $$ + $$\small \nabla \times (\mathbf{A} \times \mathbf{B}) = (\mathbf{B} \cdot \nabla) \mathbf{A} - (\mathbf{A} \cdot \nabla) \mathbf{B} + \mathbf{A} (\nabla \cdot \mathbf{B}) - \mathbf{B} (\nabla \cdot \mathbf{A}) $$ ## Sources From 820e69fc3cdd223ff020947b4330ebb5066cd471 Mon Sep 17 00:00:00 2001 From: Kevin Lin Date: Mon, 27 Jul 2026 00:32:14 -0700 Subject: [PATCH 04/16] style --- stemformulas-next/src/app/formulas/page.module.css | 4 ++-- stemformulas-next/src/app/formulas/page.tsx | 2 +- 2 files changed, 3 insertions(+), 3 deletions(-) diff --git a/stemformulas-next/src/app/formulas/page.module.css b/stemformulas-next/src/app/formulas/page.module.css index ce77c8b2..87f2b7b6 100644 --- a/stemformulas-next/src/app/formulas/page.module.css +++ b/stemformulas-next/src/app/formulas/page.module.css @@ -19,10 +19,10 @@ flex-direction: column; align-items: center; justify-content: center; - gap: 0.25rem; + gap: 0.125rem; border: 1px solid darkgray; border-radius: 12px; - padding: 0.75rem; + padding: 0.5rem 0.75rem; overflow: hidden; text-decoration: none; color: inherit; diff --git a/stemformulas-next/src/app/formulas/page.tsx b/stemformulas-next/src/app/formulas/page.tsx index 7bf40812..41a21d56 100644 --- a/stemformulas-next/src/app/formulas/page.tsx +++ b/stemformulas-next/src/app/formulas/page.tsx @@ -60,7 +60,7 @@ export default function FormulasPage() { href={`/formulas/${formula.slug}`} className={styles.card} > -

{formula.title}

+ {formula.title} {renderedLatex && (
Date: Mon, 27 Jul 2026 00:40:29 -0700 Subject: [PATCH 05/16] home page, search, more styling --- stemformulas-next/package-lock.json | 14 ++ stemformulas-next/package.json | 1 + .../src/app/api/search-index/route.ts | 6 + .../src/app/formulas/page.module.css | 2 +- stemformulas-next/src/app/globals.css | 3 +- stemformulas-next/src/app/layout.tsx | 29 +--- stemformulas-next/src/app/page.tsx | 65 +-------- stemformulas-next/src/components/Header.tsx | 80 +++++++++++ .../src/components/HomeSearch.tsx | 123 +++++++++++++++++ stemformulas-next/src/components/Search.tsx | 128 ++++++++++++++++++ .../src/components/home.module.css | 124 +++++++++++++++++ .../src/components/search.module.css | 124 +++++++++++++++++ stemformulas-next/src/lib/formulas.ts | 32 +++++ 13 files changed, 639 insertions(+), 92 deletions(-) create mode 100644 stemformulas-next/src/app/api/search-index/route.ts create mode 100644 stemformulas-next/src/components/Header.tsx create mode 100644 stemformulas-next/src/components/HomeSearch.tsx create mode 100644 stemformulas-next/src/components/Search.tsx create mode 100644 stemformulas-next/src/components/home.module.css create mode 100644 stemformulas-next/src/components/search.module.css create mode 100644 stemformulas-next/src/lib/formulas.ts diff --git a/stemformulas-next/package-lock.json b/stemformulas-next/package-lock.json index 96fb36d6..30f2e665 100644 --- a/stemformulas-next/package-lock.json +++ b/stemformulas-next/package-lock.json @@ -8,6 +8,7 @@ "name": "stemformulas-next", "version": "0.1.0", "dependencies": { + "fuse.js": "^7.5.0", "gray-matter": "^4.0.3", "next": "16.2.12", "next-mdx-remote": "^6.0.0", @@ -3942,6 +3943,19 @@ "url": "https://github.com/sponsors/ljharb" } }, + "node_modules/fuse.js": { + "version": "7.5.0", + "resolved": "https://registry.npmjs.org/fuse.js/-/fuse.js-7.5.0.tgz", + "integrity": "sha512-sQtrEfA+ez/3G0cCZecF70oqpCRttCexYUG4mUrtWL49ULUzUyxokt5kyqwtKzj1270RaKih+hcP3qLcumccow==", + "license": "Apache-2.0", + "engines": { + "node": ">=10" + }, + "funding": { + "type": "github", + "url": "https://github.com/sponsors/krisk" + } + }, "node_modules/generator-function": { "version": "2.0.1", "resolved": "https://registry.npmjs.org/generator-function/-/generator-function-2.0.1.tgz", diff --git a/stemformulas-next/package.json b/stemformulas-next/package.json index 2d9d3008..1f9fc907 100644 --- a/stemformulas-next/package.json +++ b/stemformulas-next/package.json @@ -9,6 +9,7 @@ "lint": "eslint" }, "dependencies": { + "fuse.js": "^7.5.0", "gray-matter": "^4.0.3", "next": "16.2.12", "next-mdx-remote": "^6.0.0", diff --git a/stemformulas-next/src/app/api/search-index/route.ts b/stemformulas-next/src/app/api/search-index/route.ts new file mode 100644 index 00000000..8565de94 --- /dev/null +++ b/stemformulas-next/src/app/api/search-index/route.ts @@ -0,0 +1,6 @@ +import { getAllFormulas } from "@/lib/formulas"; + +export async function GET() { + const formulas = getAllFormulas(); + return Response.json(formulas); +} diff --git a/stemformulas-next/src/app/formulas/page.module.css b/stemformulas-next/src/app/formulas/page.module.css index 87f2b7b6..0146ba7b 100644 --- a/stemformulas-next/src/app/formulas/page.module.css +++ b/stemformulas-next/src/app/formulas/page.module.css @@ -1,7 +1,7 @@ .heading { font-size: 2.25rem; font-weight: 800; - margin-top: 0; + margin: 0; } .subtitle { diff --git a/stemformulas-next/src/app/globals.css b/stemformulas-next/src/app/globals.css index 77ca32a4..3c95e3d6 100644 --- a/stemformulas-next/src/app/globals.css +++ b/stemformulas-next/src/app/globals.css @@ -73,6 +73,7 @@ a { .site-header { padding: 1.5rem 0; font-weight: 600; + font-size: 1.125rem; } @media (min-width: 640px) { @@ -166,7 +167,7 @@ a { background: none; border: none; cursor: pointer; - font-size: 1rem; + font-size: 1.125rem; color: inherit; padding: 0; } diff --git a/stemformulas-next/src/app/layout.tsx b/stemformulas-next/src/app/layout.tsx index 5c85419a..d9fed7c4 100644 --- a/stemformulas-next/src/app/layout.tsx +++ b/stemformulas-next/src/app/layout.tsx @@ -1,7 +1,7 @@ import type { Metadata } from "next"; import "./globals.css"; import "katex/dist/katex.min.css"; -import Link from "next/link"; +import Header from "@/components/Header"; export const metadata: Metadata = { title: "stemformulas", @@ -17,32 +17,7 @@ export default function RootLayout({ return ( -
- -
+
{children}
diff --git a/stemformulas-next/src/app/page.tsx b/stemformulas-next/src/app/page.tsx index 7b947a2d..0c0284d7 100644 --- a/stemformulas-next/src/app/page.tsx +++ b/stemformulas-next/src/app/page.tsx @@ -1,66 +1,5 @@ -import Image from "next/image"; -import styles from "./page.module.css"; +import HomeSearch from "@/components/HomeSearch"; export default function Home() { - return ( - - ); + return ; } diff --git a/stemformulas-next/src/components/Header.tsx b/stemformulas-next/src/components/Header.tsx new file mode 100644 index 00000000..51de9697 --- /dev/null +++ b/stemformulas-next/src/components/Header.tsx @@ -0,0 +1,80 @@ +"use client"; + +import { useEffect, useState } from "react"; +import Link from "next/link"; +import Search from "./Search"; + +export default function Header() { + const [searchOpen, setSearchOpen] = useState(false); + + useEffect(() => { + function handleKey(e: KeyboardEvent) { + if ( + e.key === "/" && + !searchOpen && + !isInputFocused() + ) { + e.preventDefault(); + setSearchOpen(true); + } + } + window.addEventListener("keydown", handleKey); + return () => window.removeEventListener("keydown", handleKey); + }, [searchOpen]); + + return ( + <> +
+ +
+ + {searchOpen && setSearchOpen(false)} />} + + ); +} + +function isInputFocused() { + const el = document.activeElement; + if (!el) return false; + const tag = el.tagName.toLowerCase(); + return ( + tag === "input" || + tag === "textarea" || + tag === "select" || + (el as HTMLElement).isContentEditable + ); +} diff --git a/stemformulas-next/src/components/HomeSearch.tsx b/stemformulas-next/src/components/HomeSearch.tsx new file mode 100644 index 00000000..915e1655 --- /dev/null +++ b/stemformulas-next/src/components/HomeSearch.tsx @@ -0,0 +1,123 @@ +"use client"; + +import { useEffect, useRef, useState } from "react"; +import { useRouter } from "next/navigation"; +import Fuse from "fuse.js"; +import type { FormulaEntry } from "@/lib/formulas"; +import styles from "./home.module.css"; + +export default function HomeSearch() { + const [results, setResults] = useState[]>([]); + const [activeIndex, setActiveIndex] = useState(-1); + const fuseRef = useRef | null>(null); + const inputRef = useRef(null); + const router = useRouter(); + + useEffect(() => { + fetch("/api/search-index") + .then((r) => r.json()) + .then((data: FormulaEntry[]) => { + fuseRef.current = new Fuse(data, { + shouldSort: true, + ignoreLocation: true, + threshold: 0.0, + includeMatches: true, + keys: [ + { name: "title", weight: 0.8 }, + { name: "summary", weight: 0.6 }, + { name: "content", weight: 0.4 }, + ], + }); + }); + }, []); + + useEffect(() => { + inputRef.current?.focus(); + }, []); + + useEffect(() => { + function handleKey(e: KeyboardEvent) { + if (e.key === "ArrowDown" && results.length > 0) { + e.preventDefault(); + setActiveIndex((i) => Math.min(i + 1, results.length - 1)); + } else if (e.key === "ArrowUp" && results.length > 0) { + e.preventDefault(); + setActiveIndex((i) => Math.max(i - 1, -1)); + } else if (e.key === "Enter" && activeIndex >= 0) { + e.preventDefault(); + router.push(results[activeIndex].item.permalink); + } else if (e.key === "Enter" && results.length > 0) { + e.preventDefault(); + router.push(results[0].item.permalink); + } + } + window.addEventListener("keydown", handleKey); + return () => window.removeEventListener("keydown", handleKey); + }, [activeIndex, results, router]); + + function handleInput(value: string) { + setActiveIndex(-1); + if (!fuseRef.current || !value.trim()) { + setResults([]); + return; + } + setResults(fuseRef.current.search(value).slice(0, 15)); + } + + return ( +
+ stemformulas logo +
+

Welcome to stemformulas.com, a modern database for STEM formulas!

+

+ Search for a formula or{" "} + submit a new one. +

+
+ +
+ ); +} diff --git a/stemformulas-next/src/components/Search.tsx b/stemformulas-next/src/components/Search.tsx new file mode 100644 index 00000000..6cbcf302 --- /dev/null +++ b/stemformulas-next/src/components/Search.tsx @@ -0,0 +1,128 @@ +"use client"; + +import { useEffect, useRef, useState } from "react"; +import { useRouter } from "next/navigation"; +import Fuse from "fuse.js"; +import type { FormulaEntry } from "@/lib/formulas"; +import styles from "./search.module.css"; + +interface Props { + onClose: () => void; +} + +export default function Search({ onClose }: Props) { + const [query, setQuery] = useState(""); + const [results, setResults] = useState[]>([]); + const [activeIndex, setActiveIndex] = useState(-1); + const fuseRef = useRef | null>(null); + const inputRef = useRef(null); + const router = useRouter(); + + useEffect(() => { + fetch("/api/search-index") + .then((r) => r.json()) + .then((data: FormulaEntry[]) => { + fuseRef.current = new Fuse(data, { + shouldSort: true, + ignoreLocation: true, + threshold: 0.0, + includeMatches: true, + keys: [ + { name: "title", weight: 0.8 }, + { name: "summary", weight: 0.6 }, + { name: "content", weight: 0.4 }, + ], + }); + }); + }, []); + + useEffect(() => { + inputRef.current?.focus(); + document.body.style.overflow = "hidden"; + return () => { + document.body.style.overflow = ""; + }; + }, []); + + function close() { + onClose(); + setQuery(""); + setResults([]); + } + + useEffect(() => { + function handleKey(e: KeyboardEvent) { + if (e.key === "Escape") { + close(); + return; + } + if (e.key === "ArrowDown") { + e.preventDefault(); + setActiveIndex((i) => Math.min(i + 1, results.length - 1)); + } else if (e.key === "ArrowUp") { + e.preventDefault(); + setActiveIndex((i) => Math.max(i - 1, -1)); + } else if (e.key === "Enter" && activeIndex >= 0) { + e.preventDefault(); + const result = results[activeIndex]; + router.push(result.item.permalink); + close(); + } + } + + window.addEventListener("keydown", handleKey); + return () => window.removeEventListener("keydown", handleKey); + }, [activeIndex, results, router, onClose]); + + function handleInput(value: string) { + setQuery(value); + setActiveIndex(-1); + if (!fuseRef.current || !value.trim()) { + setResults([]); + return; + } + const r = fuseRef.current.search(value); + setResults(r.slice(0, 15)); + } + + return ( +
+
e.stopPropagation()}> +
+ + + + + + handleInput(e.target.value)} + /> +
+
+ {results.map((r, i) => ( + + ))} +
+
+
+ ); +} diff --git a/stemformulas-next/src/components/home.module.css b/stemformulas-next/src/components/home.module.css new file mode 100644 index 00000000..951e7581 --- /dev/null +++ b/stemformulas-next/src/components/home.module.css @@ -0,0 +1,124 @@ +.wrapper { + display: flex; + flex-direction: column; + align-items: center; + padding: 2vh 1rem; + min-height: calc(100vh - 6rem); +} + +.logo { + margin-bottom: 1.5rem; +} + +.tagline { + text-align: center; + margin-bottom: 2.5rem; +} + +.tagline p { + margin: 0.25rem 0; +} + +.tagline a { + color: var(--color-primary-600); + text-decoration: underline; + font-weight: 500; + text-decoration-color: var(--color-primary-300); +} + +.tagline a:hover { + color: var(--color-neutral); + text-decoration: none; + background-color: var(--color-primary-600); + border-radius: 0.09rem; +} + +.searchBox { + display: flex; + flex-direction: column; + width: 100%; + max-width: 48rem; + min-height: 0; + margin: 0 auto; + border: 1px solid var(--color-neutral-400); + border-radius: 0.375rem; + box-shadow: 0 10px 15px -3px rgba(0, 0, 0, 0.1); + background: #fff; +} + +.searchHeader { + position: relative; + z-index: 10; + display: flex; + align-items: center; + flex: none; + padding: 0 0.5rem; +} + +.searchIcon { + display: flex; + align-items: center; + justify-content: center; + width: 2rem; + height: 2rem; + color: var(--color-neutral-400); +} + +.searchInput { + flex: auto; + min-width: 0; + height: 3rem; + margin: 0 0.25rem; + background: transparent; + border: none; + outline: none; + font-size: 1rem; + color: var(--color-neutral-900); +} + +.searchInput::placeholder { + color: var(--color-neutral-400); +} + +.resultsSection { + flex: auto; + padding: 0 0.5rem; + overflow: auto; + max-height: 50vh; +} + +.result { + display: flex; + align-items: center; + padding: 0.75rem 0.75rem; + margin-bottom: 0.5rem; + border-radius: 0.375rem; + background: var(--color-neutral-100); + text-decoration: none; + color: inherit; +} + +.result:hover, +.result.active { + background: var(--color-primary-100); +} + +.resultText { + flex-grow: 1; +} + +.resultTitle { + font-size: 1.125rem; + font-weight: 700; + margin-bottom: -0.25rem; +} + +.resultSummary { + font-size: 0.875rem; + font-style: italic; +} + +.resultArrow { + margin-left: 0.5rem; + color: var(--color-neutral-500); +} diff --git a/stemformulas-next/src/components/search.module.css b/stemformulas-next/src/components/search.module.css new file mode 100644 index 00000000..ef10a7df --- /dev/null +++ b/stemformulas-next/src/components/search.module.css @@ -0,0 +1,124 @@ +.overlay { + position: fixed; + inset: 0; + z-index: 50; + display: flex; + flex-direction: column; + cursor: default; + background: rgba(107, 114, 128, 0.5); + backdrop-filter: blur(4px); + padding: 1rem; +} + +@media (min-width: 640px) { + .overlay { + padding: 1.5rem; + } +} + +@media (min-width: 768px) { + .overlay { + padding: 10vh; + } +} + +@media (min-width: 1024px) { + .overlay { + padding: 12vh; + } +} + +.modal { + display: flex; + flex-direction: column; + width: 100%; + max-width: 48rem; + min-height: 0; + margin: 5rem auto 0; + border: 1px solid var(--color-neutral-200); + border-radius: 0.375rem; + box-shadow: 0 10px 15px -3px rgba(0, 0, 0, 0.1); + background: #fff; +} + +.header { + position: relative; + z-index: 10; + display: flex; + align-items: center; + flex: none; + padding: 0 0.5rem; +} + +.icon { + display: flex; + align-items: center; + justify-content: center; + width: 2rem; + height: 2rem; + color: var(--color-neutral-400); +} + +.input { + flex: auto; + min-width: 0; + height: 3rem; + margin: 0 0.25rem; + background: transparent; + border: none; + outline: none; + font-size: 1rem; + color: var(--color-neutral-900); +} + +.input::placeholder { + color: var(--color-neutral-400); +} + +.results { + flex: auto; + padding: 0 0.5rem; + overflow: auto; + max-height: 60vh; +} + +.result { + display: flex; + align-items: center; + width: 100%; + padding: 0.75rem 0.75rem; + margin-bottom: 0.5rem; + border-radius: 0.375rem; + border: none; + background: var(--color-neutral-100); + cursor: pointer; + text-align: left; + font-family: inherit; + font-size: inherit; + color: inherit; +} + +.result:hover, +.result.active { + background: var(--color-primary-100); +} + +.resultText { + flex-grow: 1; +} + +.resultTitle { + font-size: 1.125rem; + font-weight: 700; + margin-bottom: -0.25rem; +} + +.resultSummary { + font-size: 0.875rem; + font-style: italic; +} + +.arrow { + margin-left: 0.5rem; + color: var(--color-neutral-500); +} diff --git a/stemformulas-next/src/lib/formulas.ts b/stemformulas-next/src/lib/formulas.ts new file mode 100644 index 00000000..9453bb4a --- /dev/null +++ b/stemformulas-next/src/lib/formulas.ts @@ -0,0 +1,32 @@ +import fs from "fs"; +import path from "path"; +import matter from "gray-matter"; + +export interface FormulaEntry { + slug: string; + title: string; + summary: string; + section: string; + content: string; + permalink: string; +} + +export function getAllFormulas(): FormulaEntry[] { + const dir = path.join(process.cwd(), "src/content/formulas"); + const files = fs.readdirSync(dir).filter((f) => f.endsWith(".mdx")); + + return files.map((file) => { + const slug = file.replace(".mdx", ""); + const source = fs.readFileSync(path.join(dir, file), "utf-8"); + const { data, content } = matter(source); + + return { + slug, + title: data.title || slug, + summary: data.summary || data.description || "", + section: "formulas", + content: content.slice(0, 500), + permalink: `/formulas/${slug}`, + }; + }); +} From 3f12c47d9a07f8571c5b49a3da7c8f4ba9a48e3f Mon Sep 17 00:00:00 2001 From: Kevin Lin Date: Mon, 27 Jul 2026 00:45:09 -0700 Subject: [PATCH 06/16] about page and styling --- .../src/app/about/page.module.css | 12 +++ stemformulas-next/src/app/about/page.tsx | 86 +++++++++++++++++++ 2 files changed, 98 insertions(+) create mode 100644 stemformulas-next/src/app/about/page.module.css create mode 100644 stemformulas-next/src/app/about/page.tsx diff --git a/stemformulas-next/src/app/about/page.module.css b/stemformulas-next/src/app/about/page.module.css new file mode 100644 index 00000000..56dc4a8c --- /dev/null +++ b/stemformulas-next/src/app/about/page.module.css @@ -0,0 +1,12 @@ +.title { + font-size: 2.25rem; + font-weight: 800; + line-height: 2.5rem; + color: var(--color-neutral-900); + margin: 0 0 0.67em 0; +} + +.section { + max-width: 65ch; + margin-top: 3rem; +} diff --git a/stemformulas-next/src/app/about/page.tsx b/stemformulas-next/src/app/about/page.tsx new file mode 100644 index 00000000..3b62a677 --- /dev/null +++ b/stemformulas-next/src/app/about/page.tsx @@ -0,0 +1,86 @@ +import type { Metadata } from "next"; +import styles from "./page.module.css"; + +export const metadata: Metadata = { + title: "about · stemformulas", + description: "About page", +}; + +export default function AboutPage() { + return ( +
+
+

about

+
+
+

+ Stemformulas is a website dedicated to providing a single place to + look for STEM formulas. It has a long way to go to become that, but + the foundation is laid out. +

+

+ It was made by a few engineering students (lead by{" "} + @linguinelabs) who were + frustrated with the current state of formula searching online. This + frustration lead to the focus of the features of the site, which + include: +

+
    +
  • + The search bar being in focus on site load, so you can search for a + formula quickly +
    • +
  • + LaTeX being copyable by just clicking on it on any formula's + page +
    • +
  • The / key being used to open the search on any page
    • +
  • + Formula pages having nice preview images when shared on social + media, so people don't even need to visit the site to see the + formula +
    • +
  • The mobile-friendliness of the site
    • +
+

Tech Stack

+

+ This site is open-source on{" "} + + GitHub + + . +

+

+ It is made with the Next.js framework, rendered with MDX and KaTeX, + and styled with CSS modules. +

+

+ It is hosted on{" "} + GitHub Pages, deployed + conveniently with GitHub Actions. +

+

Contributing

+

+ If you want to add a formula to this site, there are two ways you can + do so. +

+
    +
  1. + Submit a formula on our submit page. +
    2. +
  2. + Create a pull request directly on the{" "} + + GitHub repo + + . More detailed contribution instructions can be found in the{" "} + + README + + . +
    3. +
+
+
+ ); +} From 9e87b368eff8037ca31297ca82fc07710c2760e3 Mon Sep 17 00:00:00 2001 From: Kevin Lin Date: Mon, 27 Jul 2026 00:49:52 -0700 Subject: [PATCH 07/16] working submit page --- stemformulas-next/package-lock.json | 41 ++++ stemformulas-next/package.json | 3 + stemformulas-next/src/app/api/submit/route.ts | 46 ++++ .../src/app/submit/page.module.css | 112 +++++++++ stemformulas-next/src/app/submit/page.tsx | 212 ++++++++++++++++++ stemformulas-next/src/components/Header.tsx | 3 - 6 files changed, 414 insertions(+), 3 deletions(-) create mode 100644 stemformulas-next/src/app/api/submit/route.ts create mode 100644 stemformulas-next/src/app/submit/page.module.css create mode 100644 stemformulas-next/src/app/submit/page.tsx diff --git a/stemformulas-next/package-lock.json b/stemformulas-next/package-lock.json index 30f2e665..bd17c8ad 100644 --- a/stemformulas-next/package-lock.json +++ b/stemformulas-next/package-lock.json @@ -8,8 +8,10 @@ "name": "stemformulas-next", "version": "0.1.0", "dependencies": { + "dompurify": "^3.4.12", "fuse.js": "^7.5.0", "gray-matter": "^4.0.3", + "marked": "^18.0.7", "next": "16.2.12", "next-mdx-remote": "^6.0.0", "react": "19.2.4", @@ -18,6 +20,7 @@ "remark-math": "^6.0.0" }, "devDependencies": { + "@types/dompurify": "^3.0.5", "@types/node": "^20", "@types/react": "^19", "@types/react-dom": "^19", @@ -1384,6 +1387,16 @@ "@types/ms": "*" } }, + "node_modules/@types/dompurify": { + "version": "3.0.5", + "resolved": "https://registry.npmjs.org/@types/dompurify/-/dompurify-3.0.5.tgz", + "integrity": "sha512-1Wg0g3BtQF7sSb27fJQAKck1HECM6zV1EB66j8JH9i3LCjYabJa0FSdiSgsD5K/RbrsR0SiraKacLB+T8ZVYAg==", + "dev": true, + "license": "MIT", + "dependencies": { + "@types/trusted-types": "*" + } + }, "node_modules/@types/estree": { "version": "1.0.9", "resolved": "https://registry.npmjs.org/@types/estree/-/estree-1.0.9.tgz", @@ -1478,6 +1491,13 @@ "@types/react": "^19.2.0" } }, + "node_modules/@types/trusted-types": { + "version": "2.0.7", + "resolved": "https://registry.npmjs.org/@types/trusted-types/-/trusted-types-2.0.7.tgz", + "integrity": "sha512-ScaPdn1dQczgbl0QFTeTOmVHFULt394XJgOQNoyVhZ6r2vLnMLJfBPd53SB52T/3G36VI1/g2MZaX0cwDuXsfw==", + "devOptional": true, + "license": "MIT" + }, "node_modules/@types/unist": { "version": "3.0.3", "resolved": "https://registry.npmjs.org/@types/unist/-/unist-3.0.3.tgz", @@ -2935,6 +2955,15 @@ "node": ">=0.10.0" } }, + "node_modules/dompurify": { + "version": "3.4.12", + "resolved": "https://registry.npmjs.org/dompurify/-/dompurify-3.4.12.tgz", + "integrity": "sha512-zQvGet8Z2sWbQhCmfFz/T5QWH2oBmjnqK3qvOjaqaNLrLEF912WamU+ohnTp0TCep/MFVHpdJuCZEdFOdTnEFg==", + "license": "(MPL-2.0 OR Apache-2.0)", + "optionalDependencies": { + "@types/trusted-types": "^2.0.7" + } + }, "node_modules/dunder-proto": { "version": "1.0.1", "resolved": "https://registry.npmjs.org/dunder-proto/-/dunder-proto-1.0.1.tgz", @@ -5261,6 +5290,18 @@ "url": "https://github.com/sponsors/sindresorhus" } }, + "node_modules/marked": { + "version": "18.0.7", + "resolved": "https://registry.npmjs.org/marked/-/marked-18.0.7.tgz", + "integrity": "sha512-iDVQ5ldaiKXn6b2JroX5kgRfmwgqolW7NpaEzTl1k/2Zh1njIEN9yniyLV/mOvWwtsE8OGgkjsCYvijuPk1dtA==", + "license": "MIT", + "bin": { + "marked": "bin/marked.js" + }, + "engines": { + "node": ">= 20" + } + }, "node_modules/math-intrinsics": { "version": "1.1.0", "resolved": "https://registry.npmjs.org/math-intrinsics/-/math-intrinsics-1.1.0.tgz", diff --git a/stemformulas-next/package.json b/stemformulas-next/package.json index 1f9fc907..2de0ef1d 100644 --- a/stemformulas-next/package.json +++ b/stemformulas-next/package.json @@ -9,8 +9,10 @@ "lint": "eslint" }, "dependencies": { + "dompurify": "^3.4.12", "fuse.js": "^7.5.0", "gray-matter": "^4.0.3", + "marked": "^18.0.7", "next": "16.2.12", "next-mdx-remote": "^6.0.0", "react": "19.2.4", @@ -19,6 +21,7 @@ "remark-math": "^6.0.0" }, "devDependencies": { + "@types/dompurify": "^3.0.5", "@types/node": "^20", "@types/react": "^19", "@types/react-dom": "^19", diff --git a/stemformulas-next/src/app/api/submit/route.ts b/stemformulas-next/src/app/api/submit/route.ts new file mode 100644 index 00000000..ee840f4a --- /dev/null +++ b/stemformulas-next/src/app/api/submit/route.ts @@ -0,0 +1,46 @@ +export async function POST(request: Request) { + const body = await request.json(); + const webhookUrl = process.env.DISCORD_WEBHOOK_URL; + + if (!webhookUrl) { + return Response.json({ error: "Webhook not configured" }, { status: 500 }); + } + + const res = await fetch(webhookUrl, { + method: "POST", + headers: { "Content-Type": "application/json" }, + body: JSON.stringify({ + embeds: [ + { + title: "New stemformulas.com formula suggestion", + color: 0x00ff00, + fields: [ + { name: "Title", value: body.title || "n/a", inline: true }, + { + name: "Description", + value: body.description || "n/a", + inline: true, + }, + { name: "Tags", value: body.tags || "n/a", inline: true }, + { + name: "LaTeX", + value: `\`\`\`latex\n${body.latex || "n/a"}\n\`\`\``, + inline: false, + }, + { + name: "Body", + value: `\`\`\`markdown\n${body.body || "n/a"}\n\`\`\``, + inline: false, + }, + ], + }, + ], + }), + }); + + if (!res.ok) { + return Response.json({ error: "Webhook failed" }, { status: 500 }); + } + + return Response.json({ ok: true }); +} diff --git a/stemformulas-next/src/app/submit/page.module.css b/stemformulas-next/src/app/submit/page.module.css new file mode 100644 index 00000000..a448da3f --- /dev/null +++ b/stemformulas-next/src/app/submit/page.module.css @@ -0,0 +1,112 @@ +.wrapper { + max-width: 48rem; + margin: 0 auto; + padding: 0 1rem; +} + +.header { + max-width: 65ch; + margin-bottom: 1.5rem; +} + +.title { + font-size: 2.25rem; + font-weight: 800; + line-height: 2.5rem; + color: var(--color-neutral-900); + margin: 0 0 0.5rem 0; +} + +.subtitle { + font-size: 1.125rem; + color: var(--color-neutral-500); + margin: 0 0 1.5rem 0; +} + +.formSection { + background: #fff; + border: 1px solid var(--color-neutral-200); + border-radius: 0.375rem; + box-shadow: 0 1px 3px rgba(0, 0, 0, 0.05); +} + +.form { + padding: 1.25rem; + display: flex; + flex-direction: column; +} + +.label { + display: block; + font-size: 1.125rem; + font-weight: 500; + color: var(--color-neutral-700); + margin-bottom: 0.5rem; +} + +.input, +.textarea { + width: 100%; + padding: 0.5rem 1rem; + margin-bottom: 1.5rem; + border-radius: 0.375rem; + border: 1px solid var(--color-neutral-200); + font-family: inherit; + font-size: 1rem; + outline: none; + background: #fff; + color: var(--color-neutral-900); +} + +.input:focus, +.textarea:focus { + ring: 2px solid var(--color-primary-500); + border-color: var(--color-primary-500); +} + +.textarea { + resize: vertical; +} + +.btn { + padding: 0.5rem 1.5rem; + font-size: 1rem; + font-weight: 600; + color: #fff; + background: var(--color-neutral-700); + border: none; + border-radius: 0.375rem; + cursor: pointer; + align-self: flex-start; +} + +.btn:hover { + background: var(--color-primary-600); +} + +.btn:disabled { + opacity: 0.5; + cursor: not-allowed; +} + +.previewSection { + max-width: 65ch; + margin-top: 2rem; +} + +.previewHeading { + font-size: 1.5rem; + font-weight: 700; + margin-bottom: 1.5rem; + color: var(--color-neutral-900); +} + +.previewContent { + background: #fff; + border: 1px solid var(--color-neutral-200); + border-radius: 0.5rem; + box-shadow: 0 1px 3px rgba(0, 0, 0, 0.05); + padding: 3rem; + padding-top: 5rem; + padding-bottom: 2rem; +} diff --git a/stemformulas-next/src/app/submit/page.tsx b/stemformulas-next/src/app/submit/page.tsx new file mode 100644 index 00000000..6dda469c --- /dev/null +++ b/stemformulas-next/src/app/submit/page.tsx @@ -0,0 +1,212 @@ +"use client"; + +import { useRef, useState } from "react"; +import { marked } from "marked"; +import DOMPurify from "dompurify"; +import styles from "./page.module.css"; + +declare global { + interface Window { + renderMathInElement: ( + el: HTMLElement, + opts?: Record + ) => void; + } +} + +export default function SubmitPage() { + const [title, setTitle] = useState("Gaussian/Normal Distribution"); + const [description, setDescription] = useState("The formula for the normal distribution."); + const [tags, setTags] = useState("math, statistics, probability theory"); + const [latex, setLatex] = useState( + "f(x) = \\frac{1}{\\sigma \\sqrt{2\\pi}} e^{-\\frac{1}{2}\\left(\\frac{x-\\mu}{\\sigma}\\right)^2}" + ); + const [body, setBody] = useState( + "The formula for the normal distribution as a function of the variable x is:\n$$ \\frac{1}{\\sigma \\sqrt{2\\pi}} e^{-\\frac{1}{2}\\left(\\frac{x-\\mu}{\\sigma}\\right)^2} $$\n\nWhere\n\n* $e$ is Euler's number,\n* $\\mu$ is the mean of the distribution, and\n* $\\sigma$ is the standard deviation of the distribution.\n\n## Sources\n\n- [Wikipedia](https://en.wikipedia.org/wiki/Normal_distribution)\n- [Britannica](https://www.britannica.com/topic/uniform-distribution-statistics)" + ); + const [showPreview, setShowPreview] = useState(false); + const [sending, setSending] = useState(false); + const previewRef = useRef(null); + + const tagList = tags.split(",").map((t) => t.trim()).filter(Boolean); + + function renderPreview() { + const sanitized = DOMPurify.sanitize(marked.parse(body) as string); + const tagsHTML = tagList + .map((tag) => `${tag}`) + .join(" "); + + const html = ` +
+
+

${DOMPurify.sanitize(title)}

+
${tagsHTML}
+
+
${sanitized}
+
+ `; + + setShowPreview(true); + + requestAnimationFrame(() => { + if (previewRef.current) { + previewRef.current.innerHTML = html; + if (window.renderMathInElement) { + window.renderMathInElement(previewRef.current, { + delimiters: [ + { left: "$$", right: "$$", display: true }, + { left: "$", right: "$", display: false }, + ], + }); + } + } + }); + } + + async function handleSubmit() { + try { + setSending(true); + const lastSubmit = localStorage.getItem("lastSubmissionTime"); + if (lastSubmit) { + const elapsed = Date.now() - parseInt(lastSubmit); + if (elapsed < 120000) { + alert( + `Please wait ${Math.ceil((120000 - elapsed) / 1000)} seconds before submitting again.` + ); + setSending(false); + return; + } + } + + const res = await fetch("/api/submit", { + method: "POST", + headers: { "Content-Type": "application/json" }, + body: JSON.stringify({ title, description, tags, latex, body }), + }); + + if (res.ok) { + localStorage.setItem("lastSubmissionTime", String(Date.now())); + alert("Submission sent successfully. Thank you!"); + } else { + alert("Failed to send submission. Please try again."); + } + } catch { + alert("An error occurred. Please try again."); + } finally { + setSending(false); + } + } + + return ( + <> + + - - - {{ $js := resources.Get "js/formula-suggest.js" | fingerprint }} - -{{ end }} \ No newline at end of file diff --git a/layouts/_default/taxonomy.html b/layouts/_default/taxonomy.html deleted file mode 100644 index 61307a4e..00000000 --- a/layouts/_default/taxonomy.html +++ /dev/null @@ -1,30 +0,0 @@ -{{ define "main" }} -
- {{ if .Params.showBreadcrumbs | default (.Site.Params.list.showBreadcrumbs | default false) }} - {{ partial "breadcrumbs.html" . }} - {{ end }} -

{{ lower .Title }}

-
- {{ if .Content }} -
-
- {{ .Content | emojify }} -
-
- {{ end }} - {{ $max := 0 }} - {{ range .Data.Terms }} - {{ if gt .Count $max }} - {{ $max = .Count }} - {{ end }} - {{ end }} -
-
- -
-
-{{ end }} diff --git a/layouts/_default/term.html b/layouts/_default/term.html deleted file mode 100644 index c61fdb1d..00000000 --- a/layouts/_default/term.html +++ /dev/null @@ -1,63 +0,0 @@ -{{ define "main" }} - - - - - - -
- {{ if .Params.showBreadcrumbs | default (.Site.Params.list.showBreadcrumbs | default false) }} - {{ partial "breadcrumbs.html" . }} - {{ end }} -

{{ .Title }}

-
- {{ if .Content }} -
-
- {{ .Content | emojify }} -
-
- {{ end }} -
- {{ if .Data.Pages }} -
- {{ if $.Params.groupByYear | default ($.Site.Params.list.groupByYear | default true) }} - {{ range (.Paginate (.Pages.GroupByDate "2006")).PageGroups }} -

- {{ .Key }} -

-
- {{ range .Pages }} - {{ partial "article-link.html" . }} - {{ end }} - {{ end }} - {{ else }} - {{ range (.Paginate .Pages).Pages }} - {{ partial "article-link.html" . }} - {{ end }} - {{ end }} -
- {{ partial "pagination.html" . }} - {{ else }} -
-

- {{ i18n "list.no_articles" | emojify }} -

-
- {{ end }} -
-{{ end }} diff --git a/layouts/partials/article-link.html b/layouts/partials/article-link.html deleted file mode 100644 index 79b6c640..00000000 --- a/layouts/partials/article-link.html +++ /dev/null @@ -1,74 +0,0 @@ -{{ with .Params.externalUrl }} - -{{ else }} - -{{ end}} - {{- $images := $.Resources.ByType "image" }} - {{- $thumbnail := $images.GetMatch "*thumb*" }} - {{- $feature := $images.GetMatch "*feature*" | default $thumbnail }} - {{- with $feature }} - - -
- - - -
- {{- end }} -

- {{ with .Params.externalUrl }} -
- {{ $.Title | emojify }} - - ↗ - ↖ - -
- {{ else }} - {{ .Title | emojify }} - {{ end }} - {{ if and .Draft .Site.Params.article.showDraftLabel }} -
- {{ partial "badge.html" (i18n "article.draft" | emojify) }} -
- {{ end }} - {{ if templates.Exists "partials/extend-article-link.html" }} - {{ partial "extend-article-link.html" . }} - {{ end }} -

- {{ partial "article-meta.html" . }} - {{ if .Params.showSummary | default (.Site.Params.list.showSummary | default false) }} -
- {{ .Summary | emojify }} -
- {{ end }} - diff --git a/layouts/partials/article-meta.html b/layouts/partials/article-meta.html deleted file mode 100644 index 28c1cceb..00000000 --- a/layouts/partials/article-meta.html +++ /dev/null @@ -1,76 +0,0 @@ -{{/* Determine the correct context and scope */}} -{{/* This allows for different logic depending on where the partial is called */}} -{{ $context := . }} -{{ $scope := default nil }} - -{{ if (reflect.IsMap . ) }} - {{ $context = .context }} - {{ $scope = cond (not .scope) nil .scope }} -{{ end }} - -{{ with $context }} - {{ $meta := newScratch }} - - {{/* Gather partials for this context */}} - {{ if .Params.showDate | default (.Site.Params.article.showDate | default true) }} - {{ if eq $scope "single"}} - {{ $meta.Add "partials" (slice (partial "meta/date.html" .Date)) }} - {{end}} - {{ end }} - - {{/* Output taxonomies */}} - {{ if or - (and (ne $scope "single") (.Params.showTaxonomies | default (.Site.Params.list.showTaxonomies | default (.Site.Params.article.showTaxonomies | default false)))) - (and (eq $scope "single") (.Params.showTaxonomies | default (.Site.Params.article.showTaxonomies | default false))) - }} -
- {{ range $taxonomy, $terms := .Site.Taxonomies }} - {{ if (gt (len ($context.GetTerms $taxonomy)) 0) }} - {{ range $context.GetTerms $taxonomy }} - {{ .LinkTitle }} - {{ end }} - {{ end }} - {{ end }} -
- {{ end }} - {{ if and (.Params.showDateUpdated | default (.Site.Params.list.showDateUpdated | default false)) (ne (partial "functions/date.html" .Date) (partial "functions/date.html" .Lastmod)) }} - {{ $meta.Add "partials" (slice (partial "meta/date-updated.html" .Lastmod)) }} - {{ end }} - - {{ if and (.Params.showWordCount | default (.Site.Params.article.showWordCount | default false)) (ne .WordCount 0) }} - {{ $meta.Add "partials" (slice (partial "meta/word-count.html" .)) }} - {{ end }} - - {{ if and (.Params.showReadingTime | default (.Site.Params.article.showReadingTime | default true)) (ne .ReadingTime 0) }} - {{ $meta.Add "partials" (slice (partial "meta/reading-time.html" .)) }} - {{ end }} - - {{ if and (eq $scope "single") (.Params.showEdit | default (.Site.Params.article.showEdit | default false)) }} - {{ $meta.Add "partials" (slice (partial "meta/edit.html" .)) }} - {{ end }} - - {{ if not (eq $scope "single")}} -
- {{ printf "%s" "$$" | printf "%s%s" .Params.latex | printf "%s%s" "$$" | printf "%s" }} -
-
- {{ else}} -
- {{ end }} - - {{/* Output partials */}} - {{ with ($meta.Get "partials") }} - {{ delimit . "·" }} - {{ end }} - - {{/* Output draft label */}} - {{ if and (eq $scope "single") (and .Draft .Site.Params.article.showDraftLabel) }} - {{ partial "badge.html" (i18n "article.draft" | emojify) }} - {{ end }} -
- -{{ end }} diff --git a/layouts/partials/head.html b/layouts/partials/head.html deleted file mode 100644 index 50bdfbbc..00000000 --- a/layouts/partials/head.html +++ /dev/null @@ -1,145 +0,0 @@ - - - {{ with .Site.Language.Params.htmlCode | default .Site.LanguageCode }} - - {{ end }} - - - - {{/* Title */}} - {{ if .IsHome -}} - {{ .Site.Title | emojify }} - - - - - - - - - - - - - - - {{- else -}} - {{ .Title | emojify }} · {{ .Site.Title | emojify }} - - - - - - - - - - - - - {{- end }} - - - {{/* Metadata */}} - - {{ with .Site.Params.keywords -}} - - {{- end }} - - {{ with .Site.Params.robots }} - - {{ end }} - {{ with .Params.robots }} - - {{ end }} - - {{ range .AlternativeOutputFormats -}} - {{ printf `` .Rel .MediaType.Type .RelPermalink ($.Site.Title | emojify) | safeHTML }} - {{ end -}} - {{/* Asset bundles */}} - {{ $assets := newScratch }} - {{ $cssScheme := resources.Get (printf "css/schemes/%s.css" (lower .Site.Params.colorScheme | default "congo")) }} - {{ if not $cssScheme }} - {{ $cssScheme = resources.Get "css/schemes/congo.css" }} - {{ end }} - {{ $assets.Add "css" (slice $cssScheme) }} - {{ $cssMain := resources.Get "css/compiled/main.css" }} - {{ $assets.Add "css" (slice $cssMain) }} - {{ $cssCustom := resources.Get "css/custom.css" }} - {{ if $cssCustom }} - {{ $assets.Add "css" (slice $cssCustom) }} - {{ end }} - {{ $bundleCSS := $assets.Get "css" | resources.Concat "css/main.bundle.css" | resources.Minify | resources.Fingerprint "sha512" }} - - {{ $jsAppearance := resources.Get "js/appearance.js" }} - {{ $jsAppearance = $jsAppearance | resources.Minify | resources.Fingerprint "sha512" }} - - {{ if .Site.Params.enableSearch | default false }} - {{ $jsFuse := resources.Get "lib/fuse/fuse.min.js" }} - {{ $jsSearch := resources.Get "js/search.js" }} - {{ $assets.Add "js" (slice $jsFuse $jsSearch) }} - {{ end }} - {{ if .Site.Params.enableCodeCopy | default false }} - {{ $jsCode := resources.Get "js/code.js" }} - {{ $assets.Add "js" (slice $jsCode) }} - {{ end }} - {{ if .Site.Params.rtl | default false }} - {{ $jsRTL := resources.Get "js/rtl.js" }} - {{ $assets.Add "js" (slice $jsRTL) }} - {{ end }} - {{ if $assets.Get "js" }} - {{ $bundleJS := $assets.Get "js" | resources.Concat "js/main.bundle.js" | resources.Minify | resources.Fingerprint "sha512" }} - - {{ end }} - {{/* Icons */}} - {{ if templates.Exists "partials/favicons.html" }} - {{ partialCached "favicons.html" .Site }} - {{ else }} - - - - - {{ end }} - {{/* Site Verification */}} - {{ with .Site.Params.verification.google }} - - {{ end }} - {{ with .Site.Params.verification.bing }} - - {{ end }} - {{ with .Site.Params.verification.pinterest }} - - {{ end }} - {{ with .Site.Params.verification.yandex }} - - {{ end }} - {{/* Social */}} - {{ template "_internal/opengraph.html" . }} - {{ template "_internal/twitter_cards.html" . }} - {{/* Schema */}} - {{ partial "schema.html" . }} - {{/* Me */}} - {{ with .Site.Author.name }}{{ end }} - {{ with .Site.Author.links }} - {{ range $links := . }} - {{ range $name, $url := $links }}{{ end }} - {{ end }} - {{ end }} - {{/* Vendor */}} - {{ partial "vendor.html" . }} - {{/* Analytics */}} - {{ partialCached "analytics.html" . }} - {{/* Extend head - eg. for custom analytics scripts, etc. */}} - {{ if templates.Exists "partials/extend-head.html" }} - {{ partial "extend-head.html" .Site }} - {{ end }} - - - diff --git a/layouts/partials/header/basic.html b/layouts/partials/header/basic.html deleted file mode 100644 index 2f4b1dd8..00000000 --- a/layouts/partials/header/basic.html +++ /dev/null @@ -1,39 +0,0 @@ -
- -
diff --git a/layouts/partials/home/custom.html b/layouts/partials/home/custom.html deleted file mode 100644 index 9f1d9af0..00000000 --- a/layouts/partials/home/custom.html +++ /dev/null @@ -1,39 +0,0 @@ -
- -stemformulas logo -
-

Welcome to stemformulas.com, a modern database for STEM formulas!

-

Search for a formula or submit a new one.

- -
-
-
-
-
- {{ partial "icon.html" "search" }} -
- -
-
-
-
    -
-
-
diff --git a/layouts/partials/logo.html b/layouts/partials/logo.html deleted file mode 100644 index 41821205..00000000 --- a/layouts/partials/logo.html +++ /dev/null @@ -1,33 +0,0 @@ -{{- if .Site.Params.header.logo }} - {{- $logo := resources.Get .Site.Params.header.logo }} - {{- if $logo }} - - - - {{- end }} -{{- end }} -{{- if .Site.Params.header.showTitle | default true }} - {{ .Site.Title | markdownify | emojify }} -{{- end }} - diff --git a/layouts/partials/meta/date.html b/layouts/partials/meta/date.html deleted file mode 100644 index 0c043936..00000000 --- a/layouts/partials/meta/date.html +++ /dev/null @@ -1,4 +0,0 @@ - - {{- /* Trim EOF */ -}} diff --git a/layouts/partials/search.html b/layouts/partials/search.html deleted file mode 100644 index 50e58260..00000000 --- a/layouts/partials/search.html +++ /dev/null @@ -1,48 +0,0 @@ - diff --git a/layouts/partials/vendor.html b/layouts/partials/vendor.html deleted file mode 100644 index b75dc1c9..00000000 --- a/layouts/partials/vendor.html +++ /dev/null @@ -1,39 +0,0 @@ - -{{/* Mermaid */}} -{{ if .Page.HasShortcode "mermaid" }} - {{ $mermaidLib := resources.Get "lib/mermaid/mermaid.min.js" }} - {{ $mermaidConfig := resources.Get "js/mermaid.js" }} - {{ $mermaidConfig := $mermaidConfig | resources.Minify }} - {{ $mermaidJS := slice $mermaidLib $mermaidConfig | resources.Concat "js/mermaid.bundle.js" | resources.Fingerprint "sha512" }} - -{{ end }} -{{/* Chart */}} -{{ if .Page.HasShortcode "chart" }} - {{ $chartLib := resources.Get "lib/chart/chart.umd.js" }} - {{ $chartConfig := resources.Get "js/chart.js" }} - {{ $chartConfig := $chartConfig | resources.Minify }} - {{ $chartJS := slice $chartLib $chartConfig | resources.Concat "js/chart.bundle.js" | resources.Fingerprint "sha512" }} - -{{ end }} -{{/* Katex */}} -{{ if .Page.HasShortcode "katex" }} - {{ $katexCSS := resources.Get "lib/katex/katex.min.css" }} - {{ $katexCSS := $katexCSS | resources.Fingerprint "sha512" }} - - {{ $katexJS := resources.Get "lib/katex/katex.min.js" }} - {{ $katexJS := $katexJS | resources.Fingerprint "sha512" }} - - {{ $katexRenderJS := resources.Get "lib/katex/auto-render.min.js" }} - {{ $katexRenderJS := $katexRenderJS | resources.Fingerprint "sha512" }} - - - {{ $katexFonts := resources.Match "lib/katex/fonts/*" }} - - {{ range $katexFonts }} - {{ .Publish }} - {{ end }} -{{ end }} diff --git a/stemformulas-next/mdx-components.tsx b/mdx-components.tsx similarity index 100% rename from stemformulas-next/mdx-components.tsx rename to mdx-components.tsx diff --git a/next-env.d.ts b/next-env.d.ts new file mode 100644 index 00000000..c4b7818f --- /dev/null +++ b/next-env.d.ts @@ -0,0 +1,6 @@ +/// +/// +import "./.next/dev/types/routes.d.ts"; + +// NOTE: This file should not be edited +// see https://nextjs.org/docs/app/api-reference/config/typescript for more information. diff --git a/stemformulas-next/next.config.ts b/next.config.ts similarity index 100% rename from stemformulas-next/next.config.ts rename to next.config.ts diff --git a/package-lock.json b/package-lock.json index defae028..543ff906 100644 --- a/package-lock.json +++ b/package-lock.json @@ -1,1999 +1,7895 @@ { - "name": "stemformulas.github.io", + "name": "stemformulas-next", + "version": "0.1.0", "lockfileVersion": 3, "requires": true, "packages": { "": { + "name": "stemformulas-next", + "version": "0.1.0", "dependencies": { - "@mdx-js/mdx": "^3.1.1", - "@mdx-js/react": "^3.1.1", - "@next/mdx": "^16.2.12", + "@next/third-parties": "^16.2.12", + "dompurify": "^3.4.12", + "fuse.js": "^7.5.0", "gray-matter": "^4.0.3", + "marked": "^18.0.7", + "next": "16.2.12", + "next-mdx-remote": "^6.0.0", + "react": "19.2.4", + "react-dom": "19.2.4", "rehype-katex": "^7.0.1", "remark-math": "^6.0.0" + }, + "devDependencies": { + "@types/dompurify": "^3.0.5", + "@types/node": "^20", + "@types/react": "^19", + "@types/react-dom": "^19", + "eslint": "^9", + "eslint-config-next": "16.2.12", + 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"^3.4.12", + "fuse.js": "^7.5.0", "gray-matter": "^4.0.3", + "marked": "^18.0.7", + "next": "16.2.12", + "next-mdx-remote": "^6.0.0", + "react": "19.2.4", + "react-dom": "19.2.4", "rehype-katex": "^7.0.1", "remark-math": "^6.0.0" + }, + "devDependencies": { + "@types/dompurify": "^3.0.5", + "@types/node": "^20", + "@types/react": "^19", + "@types/react-dom": "^19", + "eslint": "^9", + "eslint-config-next": "16.2.12", + "typescript": "^5" } } diff --git a/public/404.html b/public/404.html new file mode 100644 index 00000000..99631a0d --- /dev/null +++ b/public/404.html @@ -0,0 +1,7 @@ +404 Page not found · stemformulas + + +

Page Not Found 😕

Error 404

It seems that the page you've requested does not exist.

\ No newline at end of file diff --git a/static/CNAME b/public/CNAME similarity index 100% rename from static/CNAME rename to public/CNAME diff --git a/public/about/index.html b/public/about/index.html new file mode 100644 index 00000000..7db500cc --- /dev/null +++ b/public/about/index.html @@ -0,0 +1,9 @@ +about · stemformulas + + +

about

Table of Contents

Stemformulas is a website dedicated to providing a single place to look for STEM formulas.

It was made by a few engineering students who were frustrated with the current state of formula searching online: +Kevin Lin and Pranjal Biswas, both based in Vancouver, Canada.

We hope you find this site useful!

Tech Stack #

This site is made with the static website framework Hugo using the Congo theme, with many HTML/CSS/JS additions from our team.

It is hosted on GitHub Pages, deployed conveniently with GitHub Actions.

Contributing #

If you want to add a formula to this site, there are two ways you can do so. The first way is easier for you and more work for us, and the second way is the opposite:

  1. Submit all the relevant information about the formula to our Google Form (can also be filled out on our suggest page).

  2. Create a pull request on the +GitHub repo. More detailed instructions can be found in the README.

Suggestions/corrections #

If you think you’ve found a mistake in one of the formula pages or if you otherwise have some suggestions, please make a GitHub account and submit an issue here.

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categories

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a/public/formulas/amperes-law/index.html b/public/formulas/amperes-law/index.html new file mode 100644 index 00000000..cdf9d0eb --- /dev/null +++ b/public/formulas/amperes-law/index.html @@ -0,0 +1,16 @@ +Ampere's Law · stemformulas + + + + +

    Ampere's Law

    physics +electromagnetism +Maxwell's equations
    Table of Contents

    $$ \oint \vec{B} \cdot d\vec{l} = \mu_0 I $$

    Where

    • The left side is a line integral of the magnetic field \(\vec{B}\) over a closed loop \(\vec{l}\), and
    • \(\mu_0\) is the permeability of free space, and
    • \(I\) is the current enclosed by the loop.

    Ampere’s law is a formula that describes the magnetic field produced by a current-carrying wire. It says that the magneitc field created by an electric current is proportional to the magnitude of the current.

    With displacement current #

    Ampere’s law has a more general form that includes displacement current. The displacement current is proportional to the rate of change of the electric flux. This site has a good explanation of why this modification is needed. The modified form is:

    $$ \oint \vec{B} \cdot d\vec{l} = \mu_0 I + \mu_0 \epsilon_0 \frac{d\vec{\phi_E}}{dt} $$

    Where, in addition to the above definitions,

    • \(\epsilon_0\) is the permittivity of free space, and
    • \(\vec{\phi_E}\) is the electric flux intercepted by the surface spanned by the integration path of the magnetic field.

    Sources #

    \ No newline at end of file diff --git a/public/formulas/cauchys-integral-theorem/index.html b/public/formulas/cauchys-integral-theorem/index.html new file mode 100644 index 00000000..a8a2824d --- /dev/null +++ b/public/formulas/cauchys-integral-theorem/index.html @@ -0,0 +1,16 @@ +Cauchy's Integral Theorem · stemformulas + + + + +
    \ No newline at end of file diff --git a/public/formulas/convolution/index.html b/public/formulas/convolution/index.html new file mode 100644 index 00000000..1dc322e7 --- /dev/null +++ b/public/formulas/convolution/index.html @@ -0,0 +1,16 @@ +Convolution · stemformulas + + + + +
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