From 3e39baaee49e0044577dcf936d462b7db5409a97 Mon Sep 17 00:00:00 2001 From: Josh Gao Date: Thu, 26 Jun 2025 14:38:07 +0800 Subject: [PATCH] update dependencies, config and post --- _config.yml | 25 +- package-lock.json | 223 +++++++------- package.json | 8 +- .../_drafts/2025-06-26-josh-adv-math-1-2.md | 35 +++ source/_posts/2023-04-14-calculas-formulas.md | 194 ------------ source/_posts/2023-04-14-josh-adv-math-1-1.md | 285 ++++++++++++++++++ 6 files changed, 447 insertions(+), 323 deletions(-) create mode 100644 source/_drafts/2025-06-26-josh-adv-math-1-2.md delete mode 100644 source/_posts/2023-04-14-calculas-formulas.md create mode 100644 source/_posts/2023-04-14-josh-adv-math-1-1.md diff --git a/_config.yml b/_config.yml index b63745d..aafe938 100644 --- a/_config.yml +++ b/_config.yml @@ -109,9 +109,6 @@ theme: next # Deployment ## Docs: https://hexo.io/docs/one-command-deployment deploy: - - type: git - repo: josh@168.138.185.92:/home/josh/documents/josh-blog-deploy.git - branch: master - type: git repo: josh@129.153.124.210:/home/josh/documents/josh-blog-deploy.git branch: master @@ -119,12 +116,6 @@ deploy: repo: https://github.com/joshgao22/joshgao22.github.io.git branch: master -# MathJax [Redundant] -# math: -# engine: 'mathjax' -# mathjax: -# src: https://cdnjs.cloudflare.com/ajax/libs/mathjax/3.2.2/es5/tex-mml-chtml.min.js - # Local Search search: path: search.xml @@ -198,12 +189,18 @@ neat_js: related_posts: enabled: true + enable_env_name: deploy filter_threshold: 0.13 related_count: 5 weight: title: 0.1 - description: 0 - keywords: 0 - tags: 0.01 - categories: 0.05 - text: 0 \ No newline at end of file + description: 0.01 + keywords: 0.01 + tags: 0.005 + categories: 0.005 + text: 0.2 + +# config of hexo-renderer-pandoc +pandoc: + args: + - --mathjax diff --git a/package-lock.json b/package-lock.json index 1a8c0a4..b6ed291 100644 --- a/package-lock.json +++ b/package-lock.json @@ -8,7 +8,7 @@ "name": "hexo-site", "version": "0.0.0", "dependencies": { - "@next-theme/plugins": "^8.23.0", + "@next-theme/plugins": "^8.23.1", "hexo": "^7.3.0", "hexo-abbrlink": "^2.2.1", "hexo-deployer-git": "^4.0.0", @@ -24,13 +24,13 @@ "hexo-neat": "^1.0.9", "hexo-related-posts": "^1.6.2", "hexo-renderer-ejs": "^2.0.0", - "hexo-renderer-pandoc": "^0.4.0", + "hexo-renderer-pandoc": "^0.5.0", "hexo-renderer-stylus": "^3.0.1", "hexo-server": "^3.0.0", "hexo-tag-cloud": "2.1.*", "hexo-theme-landscape": "^1.1.0", - "hexo-theme-next": "^8.23.0", - "hexo-word-counter": "^0.2.0" + "hexo-theme-next": "^8.23.1", + "hexo-word-counter": "^0.2.1" } }, "node_modules/@adobe/css-tools": { @@ -39,180 +39,180 @@ "integrity": "sha512-DA5a1C0gD/pLOvhv33YMrbf2FK3oUzwNl9oOJqE4XVjuEtt6XIakRcsd7eLiOSPkp1kTRQGICTA8cKra/vFbjw==" }, "node_modules/@algolia/client-abtesting": { - "version": "5.23.0", - "resolved": "https://registry.npmjs.org/@algolia/client-abtesting/-/client-abtesting-5.23.0.tgz", - "integrity": "sha512-AyZ+9CUgWXwaaJ2lSwOJSy+/w0MFBPFqLrjWYs/HEpYMzBuFfGNZ7gEM9a7h4j7jY8hSBARBl8qdvInmj5vOEQ==", + "version": "5.20.2", + "resolved": "https://registry.npmjs.org/@algolia/client-abtesting/-/client-abtesting-5.20.2.tgz", + "integrity": "sha512-IS8JSFsDD33haaKIIFaL7qj3bEIG9GldZfb3ILW0QF3at7TcrIJYy58hrDvFee5T3p3E2aH/+wqIr0eha8jB/w==", "license": "MIT", "dependencies": { - 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"version": "0.2.0", - "resolved": "https://registry.npmjs.org/hexo-word-counter/-/hexo-word-counter-0.2.0.tgz", - "integrity": "sha512-cHFGjynprFsSrJoYEi7ar2peoavjTy7ZBDf/bQ0doXocL/ZjR42aMIcKtTi2A64KLvyTqzyb2mdKADCeclhCpQ==", + "version": "0.2.1", + "resolved": "https://registry.npmjs.org/hexo-word-counter/-/hexo-word-counter-0.2.1.tgz", + "integrity": "sha512-LkdXM9SykbVzZBGhCrYOonb/4FrNCQhVwnigt7pkVSa43Zejq3+VsG0kYmOZxDVlBmNCD1iokF6OZ3d6cYRa1Q==", "hasInstallScript": true, + "license": "LGPL-3.0", "dependencies": { "cargo-cp-artifact": "^0.1" } diff --git a/package.json b/package.json index 35571b3..c6cd520 100644 --- a/package.json +++ b/package.json @@ -12,7 +12,7 @@ "version": "7.3.0" }, "dependencies": { - "@next-theme/plugins": "^8.23.0", + "@next-theme/plugins": "^8.23.1", "hexo": "^7.3.0", "hexo-abbrlink": "^2.2.1", "hexo-deployer-git": "^4.0.0", @@ -28,12 +28,12 @@ "hexo-neat": "^1.0.9", "hexo-related-posts": "^1.6.2", "hexo-renderer-ejs": "^2.0.0", - "hexo-renderer-pandoc": "^0.4.0", + "hexo-renderer-pandoc": "^0.5.0", "hexo-renderer-stylus": "^3.0.1", "hexo-server": "^3.0.0", "hexo-tag-cloud": "2.1.*", "hexo-theme-landscape": "^1.1.0", - "hexo-theme-next": "^8.23.0", - "hexo-word-counter": "^0.2.0" + "hexo-theme-next": "^8.23.1", + "hexo-word-counter": "^0.2.1" } } \ No newline at end of file diff --git a/source/_drafts/2025-06-26-josh-adv-math-1-2.md b/source/_drafts/2025-06-26-josh-adv-math-1-2.md new file mode 100644 index 0000000..d5e48a1 --- /dev/null +++ b/source/_drafts/2025-06-26-josh-adv-math-1-2.md @@ -0,0 +1,35 @@ +--- +title: Josh's Summary — 工科数学分析
Part 1.2 常微分方程 +mathjax: true +comments: true +copyright: true +toc: + enable: true + number: true + wrap: true + expand_all: true + max_depth: 4 +tags: + - 高等数学 + - 数学分析 + - 微积分 +categories: + - - Josh 的学习笔记 + - 高等数学 + - - 数学 + - 数学分析 +abbrlink: 5a263f58 +date: 2025-06-26 10:21:02 +--- + +# 常见等价无穷小 + +1. $\sin x \sim x$ +2. $\tan x \sim x$ +3. $\arcsin x \sim x$ +4. $\arctan x \sim x$ +5. $1 - \cos x \sim \frac{x^2}{2}$ +6. $\ln(1+x) \sim x$ +7. $e^x - 1 \sim x$ +8. $a^x - 1 \sim x \ln a$ +9. $(1+x)^\alpha - 1 \sim \alpha x$ diff --git a/source/_posts/2023-04-14-calculas-formulas.md b/source/_posts/2023-04-14-calculas-formulas.md deleted file mode 100644 index bd9cc11..0000000 --- a/source/_posts/2023-04-14-calculas-formulas.md +++ /dev/null @@ -1,194 +0,0 @@ ---- -title: 高等数学常用公式 -mathjax: true -comments: true -copyright: true -toc: - enable: true - number: false - wrap: true - expand_all: true - max_depth: 4 -abbrlink: 1823c9d1 -date: 2023-04-14 16:25:59 -tags: - - 等价无穷小 - - 导数公式 - - Maclaurin 公式 - - 曲率 - - 积分公式 - - 三角函数 -categories: - - [数学, 微积分] ---- - -# 1. 常见等价无穷小 - -|||| -| :--: | :--: | :--: | -| $\sin x \sim x$ | $\tan x \sim x$ | $\arcsin x \sim x$ | -| $\arctan x \sim x$ | $\displaystyle\left( 1 - \cos x \right) \sim \frac{x^2}{2}$ | $\ln (1+x) \sim x$ | -| $\left( e^x -1 \right) \sim x$ | $\left( a^x - 1 \right) \sim x \ln a$ | $\left( \left( 1+x \right) ^\alpha -1 \right) \sim \alpha x$ | - - - -# 2. 导数基本公式 - -| 一般函数 | 三角函数 | 反三角函数 | 双曲函数
(圆函数) | -| :--: | :--: | :--: | :----: | -| $\left( C \right)' = 0$ | $\left( \sin x \right)' = \cos x$ | $\displaystyle\left( \arcsin x \right)' = \frac{1}{\sqrt{1-x^2}}$ | $\displaystyle\left( \sinh x \right)' = \left(\frac{e^x-e^{-x}}{2}\right)' = \cosh x$ | -| $\left( x^\alpha \right)'=\alpha x^{\alpha-1} \left( \alpha \in \mathbf{Z} \right)$ | $\left( \cos x \right)' = -\sin x$ | $\displaystyle\left( \arccos x \right)' = -\frac{1}{\sqrt{1-x^2}}$ | $\displaystyle\left( \cosh x \right)' = \left( \frac{e^x+e^{-x}}{2} \right)'=\sinh x$ | -| $\displaystyle\left( \log_ax\right)'=\frac{1}{x\ln a}$ | $\displaystyle\left( \tan x \right)' = \frac{1}{\cos^2 x}=\sec^2 x$ | $\displaystyle\left( \arctan x \right)' = \frac{1}{1+x^2}$ | $\displaystyle\left( \mathrm{arsinh} \, x \right)' = \left(\ln \left( x+\sqrt{x^2 + 1} \right) \right)' = \frac{1}{x^2+1}$ | -| $( \ln$\| $x$ \|$)'=\displaystyle\frac{1}{x}$ | $\displaystyle\left( \cot x \right)' = -\frac{1}{\sin^2 x} = -\csc^2 x$ | $\displaystyle\left( \mathrm{arccot} \, x \right)' = -\frac{1}{1+x^2}$ | $\displaystyle\left( \mathrm{arcosh} \, x \right)' = \left( \ln \left( x + \sqrt{x^2 - 1} \right) \right)' = \frac{1}{\sqrt{x^2-1}}$ -| | $\left( \sec x \right)' = \sec x \cdot \tan x$ | -| | $\left( \csc x \right)' = \csc x \cdot \cot x$ | - -- Newton-Leibniz 公式： - - $$ - \begin{aligned} - \left( u \cdot v \right)^{\left( n \right)} &= u^{\left( n \right)}v + nu^{\left( n-1 \right)}v' + \frac{n \left( n-1 \right)}{2!}u^{\left( n-2 \right)}v'' + \cdots + \frac{n\left( n-1 \right)\cdots\left(n-k+1\right)}{k!}u^{\left( n-k \right)}v^{(k)} + \cdots + uv^{\left( n \right)} \\ - &= \sum_{k=0}^{n}C_n^k u^{\left( u-k \right)}v^k - \end{aligned} - $$ - - 可以注意到 Newton-Leibniz 公式具有类似二项式定理展开的形式。 - -# 3. 几个初等函数的 Maclaurin 公式 - -## 3.1 Taylor 公式的一般形式 - -$$ -f(x)=f(x_0) + f'(x_0)(x-x_0) + \frac{f''(x_0)}{2!}(x-x_0)^2 + \cdots + \frac{f^{(n)}(x_0)}{n!}(x-x_0)^n + R_n -$$ - -其中 - -$$ -R_n = \frac{f^{(n+1)}(\xi)}{(n+1)!}(x-x_0)^{n+1} \, (\xi\,在\,x\,和\,x_0\,之间) = o\left( \left( x - x_0 \right)^n \right)(x\to x_0) -$$ - -## 3.2 Maclaurin 公式的一般形式 - -$$ -f(x)=f(0)+f'(0)\,x + \frac{f''(0)}{2!}x^2 + \cdots + \frac{f^{(n)}(0)}{n!}x^n + R_n -$$ - -其中 - -$$ -R_n = \frac{f^{(n+1)}(\xi)}{(n+1)!}x^{n+1} \, (\xi\,在\,0\,和\,x\,之间) = \frac{f^{(n+1)}(\theta x)}{(n+1)!} x^{n+1}(0<\theta<1) = o\left( \left( x - x_0 \right)^n \right)(x\to x_0) -$$ - -## 3.3 指数函数的 Maclaurin 展开式 - -$$ -e^x=1+x+\frac{x^2}{2} + \frac{x^3}{3} + \cdots +\frac{x^n}{n!} + \frac{e^{\theta x}}{(n+1)!}(0<\theta<1) -$$ - -## 3.4 正弦函数的 Maclaurin 展开式 - -$$ -\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots + (-1)^{m-1}\frac{x^{2m-1}}{(2m-1)!} + R_{2m} -$$ - -其中 - -$$ -R_{2m}=\frac{\sin\left( \theta x + \frac{(2m-1)\pi}{2}\right)}{(2m+1)!}x^{2m+1} = o(x^{2m})(0<\theta <1) -$$ - -## 3.5 余弦函数的 Maclaurin 展开式 - -$$ -\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} + \cdots + (-1)^m\frac{x^{2m}}{2m!} + R_{2m+1} -$$ - -其中 - -$$ -R_{2m+1}=\frac{\cos(\theta x + (m+1)\pi)}{(2m+2)!} x^{2m+2}=o(x^{2m+2}) (0<\theta <1) -$$ - -## 3.6 $\ln (1+x)$ 的 Maclaurin 展开式 - -$$ -\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} -\frac{x^4}{4} + \cdots + \frac{(-1)^{n-1}x^n}{n} + R_n(x) -$$ - -其中 - -$$ -R_n(x) = \frac{(-1)^n}{(n+1)(1+\theta x)^{n+1}}x^{n+1} = o(x^n)(0<\theta <1) -$$ - -## 3.7 $(1+x)^\alpha$ 的 Maclaurin 展开式 - -$$ -(1+x)^\alpha = 1 + \alpha x + \frac{\alpha(\alpha-1)}{2!}x^2 + \cdots + \frac{\alpha(\alpha - 1)\cdots (\alpha-n+1)}{n!}x^n + R_n(x) -$$ - -其中 - -$$ -R_n(x) = \frac{\alpha(\alpha -1)\cdots(\alpha-n)}{(n+1)!}(1+\theta x)^{\alpha-n-1}x^{n+1} = o(x)(0<\theta <1) -$$ - -## 3.8 $\frac{1}{1-x}$ 的 Maclaurin 展开式 - -$$ -\frac{1}{1-x} = 1+x+x^2+\cdots+x^n +o(x^n) -$$ - -# 4. 曲率及曲率半径 - -| 曲率 | 曲率半径 | -| :--: | :--: | -| $K=\left\| \dfrac{\mathrm{d}\alpha}{\mathrm{d}s}\right\| \xlongequal{\tan \alpha = y' \Rightarrow \alpha = \arctan y'} \dfrac{\left\| \frac{y''}{1+y'^2}\mathrm{d}x \right\|}{\sqrt{1+y'^2}}\mathrm{d}x = \dfrac{\left\| y'' \right\|}{\left( 1 + y' \right)^\frac{3}{2}}$ | $R = \frac{1}{K}$ | - -# 5. 积分常用公式 - -| 序 | 描述 | 公式 | -| :--: | :----: | :----------------------------------------------:| -| 1 | 定积分的定义 | $\displaystyle\int^1_0f(x)\mathrm{d}x=I=\lim_{\lambda \to 0}\sum_{i=1}^{n}f(\xi_i)\Delta x_i$，其中 $\lambda = \max \left\{ \Delta x_1,\Delta x_2, \cdots, \Delta x_n \right\}$ | -| 2 | 变上限积分 | $考察定积分\displaystyle\int^x_af(x)\mathrm{d}x=\int^x_af(t)\mathrm{d}t$，对 $\displaystyle\Phi(x)=\int_a^xf(t)\mathrm{d}t$，有 $\displaystyle\Phi'(x)=\frac{\mathrm{d}}{\mathrm{d}x}\int_a^xf(t)\mathrm{d}t = f(x)$ | -| 3. | — | $\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\int^{\beta(x)}_{\alpha(x)}f'(t)\mathrm{d}t = f\left[ \beta\left(x\right)\right]\beta'(x)-f[\alpha(x)]\alpha'(x)$ | -| 4 | 上下限对称的积分 | 若 $f(x)$ 为偶函数，则 $\displaystyle\int_{-a}^af(x)\mathrm{d}x=2\int_0^af(x)\mathrm{d}x$；若 $f(x)$ 为奇函数，则 $\displaystyle\int_{-a}^af(x)\mathrm{d}x=0$ | -| 5 | 周期函数的积分 | $\displaystyle f(x+T)=f(x)\Rightarrow\int_0^{nT}f(x)\mathrm{d}x=n\int^T_0f(x)\mathrm{d}x$ | -| 6 | 分部积分法 | $\displaystyle\int uv'\mathrm{d}x=uv - \int u'v\mathrm{d}x \Leftrightarrow \int u\mathrm{d}v = uv - \int v\mathrm{d}u$ | - -# 6. 基本积分表 - -| 序 | 积分式 | -| :--: | :-------------------------: | -| 1 | $\displaystyle\int k\mathrm{d}x=kx+C$，特别地 $\displaystyle\int\mathrm{d}x=x+C$ | -| 2 | $\displaystyle\int x^\mu\mathrm{d}x=\frac{x^{\mu+1}}{\mu+1}+C(\mu\ne-1)$，特别地 $\displaystyle\int \frac{\mathrm{d}x}{x^2}=-\frac{1}{x}+C,\int\frac{\mathrm{d}x}{\sqrt{x}}=2\sqrt{x}+C$ | -| 3 | $\displaystyle\int\frac{\mathrm{d}x}{x}=\ln \|x\|+C$ | -| 4 | $\displaystyle\int\frac{\mathrm{d}x}{1+x^2} = \arctan x +C$ | -| 5 | $\displaystyle\int\frac{\mathrm{d}x}{\sqrt{1-x^2}}=\arcsin x+C$ | -| 6 | $\displaystyle\int\sin x\mathrm{d}x = -\cos x+C, \int\cos x\mathrm{d}x = \sin x +C$ | -| 7 | $\displaystyle\int\frac{\mathrm{d}x}{\cos^2x} = \int\sec^2x\mathrm{d}x = \tan x +C, \int\frac{\mathrm{d}x}{\sin^2x}=\int\csc^2x\mathrm{d}x = -\cot x +C$ | -| 8 | $\displaystyle\int a^x\mathrm{d}x = \frac{a^x}{\ln a} + C$，特别地 $\displaystyle\int e^x\mathrm{d}x = e^x +C$ -| 9 | $\displaystyle\int \sinh x\mathrm{d}x = \cosh x + C，\int \cosh x \mathrm{d}x=\sinh x +C$ | - -# 7. 三角函数相关公式 - -| 序 | 公式 | -| :--: | :-------------------------: | -| 1 | $\sin^2x+\cos^2x=1, \tan^2x+1=\sec^2x,\cot^2x+1=\csc^2x$ | -| 2 | $\begin{aligned}&\cosh^2x-\sinh^2x=1,\\ &\sinh(x\pm y)=\sinh x\cdot\cosh y \pm \sinh y\cdot\cosh x, \cosh(x\pm y) = \cosh x\cdot\cosh y,\\ &\sinh2x = 2\sinh x\cdot \cosh x, \cosh 2x = \cosh^2x+\sinh^2x = 2\cosh^2x - 1\end{aligned}$ | -| 3
积化和差公式 | $\sin\alpha\cos\beta=\frac12[\sin(\alpha+\beta)+\sin(\alpha-\beta)],\cos\alpha\sin\beta = \frac12[\sin(\alpha+\beta) - \sin(\alpha-\beta)]$
$\cos\alpha\cos\beta = \frac12[cos(\alpha+\beta) + cos(\alpha - \beta)], \sin\alpha\sin\beta=-\frac12[\cos(\alpha+\beta) - \cos(\alpha-\beta)]$ | -| 4
和差化积公式 | $\displaystyle\sin\alpha + \sin\beta = 2\sin\left(\frac{\alpha+\beta}{2}\right) \cos\left(\frac{\alpha-\beta}{2}\right), \sin\alpha - \sin\beta = 2\sin\left(\frac{\alpha-\beta}{2}\right) \cos\left(\frac{\alpha+\beta}{2}\right)$
$\displaystyle\cos\alpha + \cos\beta = 2\cos\left(\frac{\alpha+\beta}{2}\right) \cos\left(\frac{\alpha-\beta}{2}\right), \cos\alpha - \cos\beta = -2\sin\left(\frac{\alpha+\beta}{2}\right) \sin\left(\frac{\alpha-\beta}{2}\right)$ | -| 5
三角代换 | $\displaystyle\sqrt{a^2-x^2}\xrightarrow{x=a\sin t}a\cos t, \sqrt{a^2+x^2} \xrightarrow{x=a\tan t}a\sec t, \sqrt{x^2-a^2}\xrightarrow{x=a\sec t}a\tan t \left( a>0,t\in\left( -\frac{\pi}{2} , \frac{\pi}{2} \right) \right)$ | -| 6 | $\displaystyle\int\sqrt{a^2-x^2}\mathrm{d}x = \frac{a^2}{2}\arcsin\frac xa + \frac12x\sqrt{a^2-x^2} + C(a>0)$ | -| 7 | $\displaystyle\int\sqrt{x^2+a^2}\mathrm{d}x = \frac x2\sqrt{x^2+a^2} + \frac{a^2}2\ln \left\| x+\sqrt{x^2+a^2} \right\|+C(a>0)$ | -| 8 | $\displaystyle\int\frac{1}{\sqrt{x^2+a^2}}\mathrm{d}x = \ln\left\| x+ \sqrt{x^2+a^2} \right\| +C$, $\displaystyle\int\frac{1}{\sqrt{x^2-a^2}}\mathrm{d}x = \ln\left\| x + \sqrt{x^2+a^2}\right\|(a>0)$，特别地，$\displaystyle\int\frac{1}{\sqrt{x^2+1}}\mathrm{d}x=\left(x+\sqrt{x^2+1}\right) = \mathrm{arsinh}\, x$, $\displaystyle\int\frac{1}{\sqrt{x^2-1}}\mathrm{d}x = \ln\left( x+\sqrt{x^2-1}\right) = \mathrm{arcosh}\, x$ | -| 9 | $\displaystyle\int\sec x\mathrm{d}x = \int\frac{1}{\cos x}\mathrm{d}x = \int\frac{\sec^2x+\sec x \tan x}{\sec x + \tan x}\mathrm{d}x = \int\frac{1}{\sec x + \tan x}\mathrm{d}(\sec x + \tan x) = \ln \left\| \sec x + \tan x \right\| + C$ | -| 10 | $\displaystyle\int\csc x\mathrm{d}x = \int\frac{1}{\sin x}\mathrm{d}x = \ln\left\| \tan\frac x2 \right\| + C = \frac 12 \ln \left\| \frac{1-\cos x}{1 + \cos x}\right\| + C$ | -| 11 | $\displaystyle\int\tan x\mathrm{d}x = \int\frac{\sin x}{\cos x}\mathrm{d}x = -\int\frac{1}{\cos x}\mathrm{d}(\cos x) = -\ln \left\| \cos x\right\| +C$ | -| 12 | $\displaystyle\int\frac{1}{\tan x}\mathrm{d}x = \int\frac{\cos x}{\sin x}\mathrm{d}x = \int\frac{1}{\sin x}\mathrm{d}(\sin x) = \ln\left\| \sin x \right\| + C$ | -| 13 | $\displaystyle\int\arctan x\mathrm{d}x = x\arctan x - \ln \left\| 1+x^2 \right\| +C$ | -| 14
Wallis 公式 | $\displaystyle\int^{\frac{\pi}{2}}_{0}\sin^nx\mathrm{d}x = \int^{\frac{\pi}{2}}_{0}\cos^nx\mathrm{d}x = \begin{cases} \displaystyle\frac{n-1}{n}\cdot\frac{n-3}{n-2}\cdot\cdots\cdot\frac12\cdot\frac\pi2,&n为偶数 \\ \displaystyle\frac{n-1}{n}\cdot\frac{n-3}{n-2}\cdot\cdots\cdot\frac23\cdot1,&n为奇数 \end{cases} \\ = \begin{cases} \displaystyle\frac{(n-1)!!}{n!!}\cdot\frac\pi2,&n为偶数 \\ \displaystyle\frac{(n-1)!!}{n!!},&n为奇数 \end{cases}$ | -| 15 | $\displaystyle\int_0^\pi\sin^nx\mathrm{d}x = 2\int^\frac\pi2_0\sin^nx\mathrm{d}x,\int_0^\pi\cos^nx\mathrm{d}x = \begin{cases} 2\displaystyle\int^\frac\pi2_0\cos^nx\mathrm{d}x,&n为偶数 \\ 0,&n为奇数 \end{cases}$
$\displaystyle\int_0^{2\pi}\sin^nx = \int_0^{2\pi}\cos^nx\mathrm{d}x = \begin{cases} 4\displaystyle\int^\frac\pi2_0\sin^nx\mathrm{d}x,&n为偶数 \\ 0,&n为奇数 \end{cases}$
$\displaystyle\int_0^\frac\pi2f(\sin x)\mathrm{d}x = \int_0^\frac\pi2f(\cos x)\mathrm{d}x$, $\displaystyle\int_0^\pi f(\sin x)\mathrm{d}x \ne \int_0^\pi f(\cos x)\mathrm{d}x$
$\displaystyle\int_0^\pi\pi f(\sin x)\mathrm{d}x = \frac\pi2\int_0^\pi f(\sin x)\mathrm{d}x = \pi\int_0^\frac\pi2 f(\sin x)\mathrm{d}x$ | -| 16
万能公式 | 令 $u=\displaystyle\tan\frac x2,x=2\arctan u$，则 $\displaystyle\sin x = \frac{2u}{1+u^2},\cos x=\frac{1-u^2}{1+u^2}, \mathrm{d}x = \frac{2}{1+u^2}\mathrm{d}u$ | -| 17 | $\displaystyle\int \frac{a\cos x+ b\sin x}{c\cos x + d \sin x} = A\int\mathrm{d}x+B\int\frac{\mathrm{d}(c\cos x+d \sin x)}{c \cos x + d \sin x}$
(设 $a\cos x + b\sin x = A(c \cos x + d \sin x) + B(c \cos x + d \sin x)'$，由待定系数法求出A、B)$ | diff --git a/source/_posts/2023-04-14-josh-adv-math-1-1.md b/source/_posts/2023-04-14-josh-adv-math-1-1.md new file mode 100644 index 0000000..a01f91f --- /dev/null +++ b/source/_posts/2023-04-14-josh-adv-math-1-1.md @@ -0,0 +1,285 @@ +--- +title: Josh's Summary — 工科数学分析
Part 1.1 基本微分与积分 +mathjax: true +comments: true +copyright: true +toc: + enable: true + number: false + wrap: true + expand_all: true + max_depth: 4 +abbrlink: 1823c9d1 +date: 2023-04-14 16:25:59 +tags: + - 等价无穷小 + - 导数公式 + - Maclaurin 公式 + - 曲率 + - 积分公式 + - 三角函数 +categories: + - - Josh 的学习笔记 + - 微积分 + - - 数学 + - 微积分 +--- + +$\newcommand\diff{\mathop{}\!\mathrm{d}}$ + +# 1. 常见等价无穷小 + +|||| +| :--: | :--: | :--: | +| $\sin x \sim x$ | $\tan x \sim x$ | $\arcsin x \sim x$ | +| $\arctan x \sim x$ | $\displaystyle\left( 1 - \cos x \right) \sim \dfrac{x^2}{2}$ | $\ln (1+x) \sim x$ | +| $\left( e^x -1 \right) \sim x$ | $\left( a^x - 1 \right) \sim x \ln a$ | $\left( \left( 1+x \right) ^\alpha -1 \right) \sim \alpha x$ | + + + +# 2. 导数基本公式 + +| 一般函数 | 三角函数 | 反三角函数 | 双曲函数
(圆函数) | +| :--: | :--: | :--: | :----: | +| $\left( C \right)' = 0$ | $\left( \sin x \right)' = \cos x$ | $\displaystyle\left( \arcsin x \right)' = \dfrac{1}{\sqrt{1-x^2}}$ | $\displaystyle\left( \sinh x \right)' = \left(\dfrac{e^x-e^{-x}}{2}\right)' = \cosh x$ | +| $\left( x^\alpha \right)'=\alpha x^{\alpha-1} \left( \alpha \in \mathbb{Z} \right)$ | $\left( \cos x \right)' = -\sin x$ | $\displaystyle\left( \arccos x \right)' = -\dfrac{1}{\sqrt{1-x^2}}$ | $\displaystyle\left( \cosh x \right)' = \left( \dfrac{e^x+e^{-x}}{2} \right)'=\sinh x$ | +| $\displaystyle\left( \log_ax\right)'=\dfrac{1}{x\ln a}$ | $\displaystyle\left( \tan x \right)' = \dfrac{1}{\cos^2 x}=\sec^2 x$ | $\displaystyle\left( \arctan x \right)' = \dfrac{1}{1+x^2}$ | $\displaystyle\left( \mathrm{arsinh} \, x \right)' = \left(\ln \left( x+\sqrt{x^2 + 1} \right) \right)' = \dfrac{1}{x^2+1}$ | +| $( \ln$\| $x$ \|$)'=\displaystyle\dfrac{1}{x}$ | $\displaystyle\left( \cot x \right)' = -\dfrac{1}{\sin^2 x} = -\csc^2 x$ | $\displaystyle\left( \mathrm{arccot} \, x \right)' = -\dfrac{1}{1+x^2}$ | $\displaystyle\left( \mathrm{arcosh} \, x \right)' = \left( \ln \left( x + \sqrt{x^2 - 1} \right) \right)' = \dfrac{1}{\sqrt{x^2-1}}$ +| | $\left( \sec x \right)' = \sec x \cdot \tan x$ | +| | $\left( \csc x \right)' = \csc x \cdot \cot x$ | + +- Newton-Leibniz 公式： + + $$ + \begin{aligned} + \left( u \cdot v \right)^{\left( n \right)} &= u^{\left( n \right)}v + nu^{\left( n-1 \right)}v' + \dfrac{n \left( n-1 \right)}{2!}u^{\left( n-2 \right)}v'' + \cdots + \dfrac{n\left( n-1 \right)\cdots\left(n-k+1\right)}{k!}u^{\left( n-k \right)}v^{(k)} + \cdots + uv^{\left( n \right)} \\ + &= \sum_{k=0}^{n}C_n^k u^{\left( u-k \right)}v^k + \end{aligned} + $$ + + 可以注意到 Newton-Leibniz 公式具有类似二项式定理展开的形式。 + +# 3. 几个初等函数的 Maclaurin 公式 + +## 3.1 Taylor 公式的一般形式 + +$$\begin{aligned} +f(x)&=f(x_0) + f'(x_0)(x-x_0) + \dfrac{f''(x_0)}{2!}(x-x_0)^2 + \cdots + \dfrac{f^{(n)}(x_0)}{n!}(x-x_0)^n + R_n +\\ +&= \sum_{k=0}^n \dfrac{f^{(n)}(x_0)}{n!}(x - x_0)^n + R_n +\end{aligned}$$ + +其中 + +$$ +R_n = \dfrac{f^{(n+1)}(\xi)}{(n+1)!}(x-x_0)^{n+1} \xlongequal{x\to x_0} o\left( \left( x - x_0 \right)^n \right), \quad \xi\,在\,x\,和\,x_0\,之间 +$$ + +## 3.2 Maclaurin 公式的一般形式 + +$$\begin{aligned} +f(x) &= f(0)+f'(0)\,x + \dfrac{f''(0)}{2!}x^2 + \cdots + \dfrac{f^{(n)}(0)}{n!}x^n + R_n +\\ +&= \sum_{k=0}^n \dfrac{f^{(n)}(0)}{n!}x^n + R_n +\end{aligned}$$ + +其中 + +$$ +R_n = \dfrac{f^{(n+1)}(\xi)}{(n+1)!}x^{n+1} = \dfrac{f^{(n+1)}(\theta x)}{(n+1)!} x^{n+1} \xlongequal{x\to x_0} o\left( \left( x - x_0 \right)^n \right), \quad \xi\,在\,0\,和\,x\,之间, 0<\theta<1 +$$ + +## 3.3 指数函数的 Maclaurin 展开式 + +$$ +e^x=1+x+\dfrac{x^2}{2} + \dfrac{x^3}{3} + \cdots +\dfrac{x^n}{n!} + \dfrac{e^{\theta x}}{(n+1)!}, \quad 0<\theta<1 +$$ + +## 3.4 正弦函数的 Maclaurin 展开式 + +$$ +\sin x = x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} + \cdots + (-1)^{m-1}\dfrac{x^{2m-1}}{(2m-1)!} + R_{2m} +$$ + +其中 + +$$ +R_{2m}=\dfrac{\sin\left( \theta x + \dfrac{(2m-1)\pi}{2}\right)}{(2m+1)!}x^{2m+1} = o(x^{2m}), \quad 0<\theta<1 +$$ + +## 3.5 余弦函数的 Maclaurin 展开式 + +$$ +\cos x = 1 - \dfrac{x^2}{2!} + \dfrac{x^4}{4!} + \cdots + (-1)^m\dfrac{x^{2m}}{2m!} + R_{2m+1} +$$ + +其中 + +$$ +R_{2m+1}=\dfrac{\cos(\theta x + (m+1)\pi)}{(2m+2)!} x^{2m+2}=o(x^{2m+2}), \quad 0<\theta<1 +$$ + +## 3.6 $\ln (1+x)$ 的 Maclaurin 展开式 + +$$ +\ln(1+x) = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} -\dfrac{x^4}{4} + \cdots + \dfrac{(-1)^{n-1}x^n}{n} + R_n(x) +$$ + +其中 + +$$ +R_n(x) = \dfrac{(-1)^n}{(n+1)(1+\theta x)^{n+1}}x^{n+1} = o(x^n), \quad 0<\theta<1 +$$ + +## 3.7 $(1+x)^\alpha$ 的 Maclaurin 展开式 + +$$ +(1+x)^\alpha = 1 + \alpha x + \dfrac{\alpha(\alpha-1)}{2!}x^2 + \cdots + \dfrac{\alpha(\alpha - 1)\cdots (\alpha-n+1)}{n!}x^n + R_n(x) +$$ + +其中 + +$$ +R_n(x) = \dfrac{\alpha(\alpha -1)\cdots(\alpha-n)}{(n+1)!}(1+\theta x)^{\alpha-n-1}x^{n+1} = o(x), \quad 0<\theta<1 +$$ + +## 3.8 $\dfrac{1}{1-x}$ 的 Maclaurin 展开式 + +$$ +\dfrac{1}{1-x} = 1+x+x^2+\cdots+x^n +o(x^n) +$$ + +# 4. 曲率及曲率半径 + +| 曲率 | 曲率半径 | +| :--: | :--: | +| $K=\left\| \dfrac{\diff \alpha}{\diff s}\right\| \xlongequal{令\ \tan \alpha = y', 则\ \alpha = \arctan y'} \dfrac{\left\| \dfrac{y''}{1+y'^2}\diff x \right\|}{\sqrt{1+y'^2}}\diff x = \dfrac{\left\| y'' \right\|}{\left( 1 + y' \right)^{3/2}}$ | $R = \dfrac{1}{K}$ | + +# 5. 积分常用公式 + +| 序 | 描述 | 公式 | +| :--: | :----: | :----------------------------------------------:| +| 1 | 定积分的定义 | $\displaystyle\int^1_0f(x)\diff x=I=\lim_{\lambda \to 0}\sum_{i=1}^{n}f(\xi_i)\Delta x_i$，其中 $\lambda = \max \left\{ \Delta x_1,\Delta x_2, \cdots, \Delta x_n \right\}$ | +| 2 | 变上限积分 | $考察定积分\displaystyle\int^x_af(x)\diff x=\int^x_af(t)\diff t$，对 $\displaystyle\Phi(x)=\int_a^xf(t)\diff t$，有 $\displaystyle\Phi'(x)=\dfrac{\diff }{\diff x}\int_a^xf(t)\diff t = f(x)$ | +| 3 | 变上限积分的扩展 | $\displaystyle\dfrac{\diff }{\diff x}\int^{\beta(x)}_{\alpha(x)}f'(t)\diff t = f\left[ \beta\left(x\right)\right]\beta'(x)-f[\alpha(x)]\alpha'(x)$ | +| 4 | 上下限对称的积分 | 若 $f(x)$ 为偶函数，则 $\displaystyle\int_{-a}^af(x)\diff x=2\int_0^af(x)\diff x$；若 $f(x)$ 为奇函数，则 $\displaystyle\int_{-a}^af(x)\diff x=0$ | +| 5 | 周期函数的积分 | $\displaystyle f(x+T)=f(x)\Rightarrow\int_0^{nT}f(x)\diff x=n\int^T_0f(x)\diff x$ | +| 6 | 分部积分法 | $\displaystyle\int uv'\diff x=uv - \int u'v\diff x \Leftrightarrow \int u\diff v = uv - \int v\diff u$ | + +# 6. 基本积分表 + +| 序 | 积分式 | +| :--: | :-------------------------: | +| 1 | $\displaystyle\int k\diff x=kx+C$，特别地 $\displaystyle\int\diff x=x+C$ | +| 2 | $\displaystyle\int x^\mu\diff x=\dfrac{x^{\mu+1}}{\mu+1}+C(\mu\ne-1)$，特别地 $\displaystyle\int \dfrac{1}{x^2}\diff x=-\dfrac{1}{x}+C,\int\dfrac{1}{\sqrt{x}}\diff x=2\sqrt{x}+C$ | +| 3 | $\displaystyle\int\dfrac{1}{x}\diff x=\ln \|x\|+C$ | +| 4 | $\displaystyle\int\dfrac{1}{1+x^2}\diff x = \arctan x +C$ | +| 5 | $\displaystyle\int\dfrac{1}{\sqrt{1-x^2}}\diff x=\arcsin x+C$ | +| 6 | $\displaystyle\int\sin x\diff x = -\cos x+C, \int\cos x\diff x = \sin x +C$ | +| 7 | $\displaystyle\int\dfrac{\diff x}{\cos^2x} = \int\sec^2x\diff x = \tan x +C, \int\dfrac{\diff x}{\sin^2x}=\int\csc^2x\diff x = -\cot x +C$ | +| 8 | $\displaystyle\int a^x\diff x = \dfrac{a^x}{\ln a} + C$，特别地 $\displaystyle\int e^x\diff x = e^x +C$ +| 9 | $\displaystyle\int \sinh x\diff x = \cosh x + C，\int \cosh x \diff x=\sinh x +C$ | + +# 7. 三角函数相关公式 + +| 序 | 公式 | +| :--: | :-------------------------: | +| 1 | $\sin^2x+\cos^2x=1, \tan^2x+1=\sec^2x,\cot^2x+1=\csc^2x$ | +| 2 | $\begin{aligned}&\cosh^2x-\sinh^2x=1,\\ &\sinh(x\pm y)=\sinh x\cdot\cosh y \pm \sinh y\cdot\cosh x, \cosh(x\pm y) = \cosh x\cdot\cosh y,\\ &\sinh2x = 2\sinh x\cdot \cosh x, \cosh 2x = \cosh^2x+\sinh^2x = 2\cosh^2x - 1\end{aligned}$ | +| 3
积化和差公式 | $\sin\alpha\cos\beta=\dfrac12[\sin(\alpha+\beta)+\sin(\alpha-\beta)],\cos\alpha\sin\beta = \dfrac12[\sin(\alpha+\beta) - \sin(\alpha-\beta)]$
$\cos\alpha\cos\beta = \dfrac12[cos(\alpha+\beta) + cos(\alpha - \beta)], \sin\alpha\sin\beta=-\dfrac12[\cos(\alpha+\beta) - \cos(\alpha-\beta)]$ | +| 4
和差化积公式 | $\displaystyle\sin\alpha + \sin\beta = 2\sin\left(\dfrac{\alpha+\beta}{2}\right) \cos\left(\dfrac{\alpha-\beta}{2}\right), \sin\alpha - \sin\beta = 2\sin\left(\dfrac{\alpha-\beta}{2}\right) \cos\left(\dfrac{\alpha+\beta}{2}\right)$
$\displaystyle\cos\alpha + \cos\beta = 2\cos\left(\dfrac{\alpha+\beta}{2}\right) \cos\left(\dfrac{\alpha-\beta}{2}\right), \cos\alpha - \cos\beta = -2\sin\left(\dfrac{\alpha+\beta}{2}\right) \sin\left(\dfrac{\alpha-\beta}{2}\right)$ | +| 5
三角代换 | $\displaystyle\sqrt{a^2-x^2}\xrightarrow{x=a\sin t}a\cos t, \sqrt{a^2+x^2} \xrightarrow{x=a\tan t}a\sec t, \sqrt{x^2-a^2}\xrightarrow{x=a\sec t}a\tan t \left( a>0,t\in\left( -\dfrac{\pi}{2} , \dfrac{\pi}{2} \right) \right)$ | +| 6 | $\displaystyle\int\sqrt{a^2-x^2}\diff x = \dfrac{a^2}{2}\arcsin\dfrac xa + \dfrac12x\sqrt{a^2-x^2} + C(a>0)$ | +| 7 | $\displaystyle\int\sqrt{x^2+a^2}\diff x = \dfrac x2\sqrt{x^2+a^2} + \dfrac{a^2}2\ln \left\| x+\sqrt{x^2+a^2} \right\|+C(a>0)$ | +| 8 | $\displaystyle\int\dfrac{1}{\sqrt{x^2+a^2}}\diff x = \ln\left\| x+ \sqrt{x^2+a^2} \right\| +C$, $\displaystyle\int\dfrac{1}{\sqrt{x^2-a^2}}\diff x = \ln\left\| x + \sqrt{x^2-a^2}\right\|(a>0)$
特别地，$\displaystyle\int\dfrac{1}{\sqrt{x^2+1}}\diff x=\ln\left(x+\sqrt{x^2+1}\right) = \mathrm{arsinh}\, x$, $\displaystyle\int\dfrac{1}{\sqrt{x^2-1}}\diff x = \ln\left( x+\sqrt{x^2-1}\right) = \mathrm{arcosh}\, x$ | +| 9 | $\displaystyle\int\sec x\diff x = \int\dfrac{1}{\cos x}\diff x = \int\dfrac{\sec^2x+\sec x \tan x}{\sec x + \tan x}\diff x = \int\dfrac{1}{\sec x + \tan x}\diff (\sec x + \tan x) = \ln \left\| \sec x + \tan x \right\| + C$ | +| 10 | $\displaystyle\int\csc x\diff x = \int\dfrac{1}{\sin x}\diff x = \ln\left\| \tan\dfrac x2 \right\| + C = \dfrac 12 \ln \left\| \dfrac{1-\cos x}{1 + \cos x}\right\| + C$ | +| 11 | $\displaystyle\int\tan x\diff x = \int\dfrac{\sin x}{\cos x}\diff x = -\int\dfrac{1}{\cos x}\diff (\cos x) = -\ln \left\| \cos x\right\| +C$ | +| 12 | $\displaystyle\int\dfrac{1}{\tan x}\diff x = \int\dfrac{\cos x}{\sin x}\diff x = \int\dfrac{1}{\sin x}\diff (\sin x) = \ln\left\| \sin x \right\| + C$ | +| 13 | $\displaystyle\int\arctan x\diff x = x\arctan x - \ln \left\| 1+x^2 \right\| +C$ | +| 14
Wallis 公式 | $\displaystyle\int^{\pi/2}_{0}\sin^nx\diff x = \int^{\pi/2}_{0}\cos^nx\diff x = \begin{cases} \displaystyle\dfrac{n-1}{n}\cdot\dfrac{n-3}{n-2}\cdot\cdots\cdot\dfrac12\cdot\dfrac\pi2,&n为偶数 \\ \displaystyle\dfrac{n-1}{n}\cdot\dfrac{n-3}{n-2}\cdot\cdots\cdot\dfrac23\cdot1,&n为奇数 \end{cases} \\ = \begin{cases} \displaystyle\dfrac{(n-1)!!}{n!!}\cdot\dfrac\pi2,&n为偶数 \\ \displaystyle\dfrac{(n-1)!!}{n!!},&n为奇数 \end{cases}$ | +| 15 | $\displaystyle\int_0^\pi\sin^nx\diff x = 2\int^{\pi/2}_0\sin^nx\diff x,\int_0^\pi\cos^nx\diff x = \begin{cases} 2\displaystyle\int^{\pi/2}_0\cos^nx\diff x,&n为偶数 \\ 0,&n为奇数 \end{cases}$
$\displaystyle\int_0^{2\pi}\sin^nx = \int_0^{2\pi}\cos^nx\diff x = \begin{cases} 4\displaystyle\int^{\pi/2}_0\sin^nx\diff x,&n为偶数 \\ 0,&n为奇数 \end{cases}$
$\displaystyle\int_0^{\pi/2}f(\sin x)\diff x = \int_0^{\pi/2}f(\cos x)\diff x$, $\displaystyle\int_0^\pi f(\sin x)\diff x \ne \int_0^\pi f(\cos x)\diff x$
$\displaystyle\int_0^\pi\pi f(\sin x)\diff x = \dfrac\pi2\int_0^\pi f(\sin x)\diff x = \pi\int_0^{\pi/2} f(\sin x)\diff x$ | +| 16
万能公式 | 令\ $u=\displaystyle\tan\dfrac x2,x=2\arctan u$，则\ $\displaystyle\sin x = \dfrac{2u}{1+u^2},\cos x=\dfrac{1-u^2}{1+u^2}, \diff x = \dfrac{2}{1+u^2}\diff u$ | +| 17 | $\displaystyle\int \dfrac{a\cos x+ b\sin x}{c\cos x + d \sin x} = A\int\diff x+B\int\dfrac{\diff (c\cos x+d \sin x)}{c \cos x + d \sin x}$
(设 $a\cos x + b\sin x = A(c \cos x + d \sin x) + B(c \cos x + d \sin x)'$，由待定系数法求出 $A, B$) | + +# 8. 定积分的几何应用 + +## 8.1 平面 + +### 8.1.1 直角坐标 + +1. 直角梯形的面积 + +$$\begin{equation*} +A = \int_a^b [f(x_1) - f(x_2)] \diff x +\end{equation*}$$ + +2. 曲边梯形的面积：对于参数方程 $\begin{cases} + x = \varphi(x) \\ y = {\psi(x)} + \end{cases}$，曲边梯形的面积（变换上下限） + + $$\begin{equation*} + A = \int_a^b y\diff x = \int_{t_1}^{t_2} \psi(t) \diff \varphi(t) = \int_{t_1}^{t_2} \psi(t)\varphi'(t) \diff t + \end{equation*}$$ + +3. 椭圆的面积：椭圆 $\begin{cases} + x = a \cos t \\ y = b \sin t + \end{cases}$，即 $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ 的面积 + + $$\begin{equation*} + A = 4 \int_0^a y \diff x = 4ab \int_0^{\pi/2} (1-\cos t)^2 \diff t = 3\pi a^2 + \end{equation*}$$ + +4. 摆线的面积：摆线 $\begin{cases} + x = a (t - \sin t) \\ y = a (1 - \cos t) + \end{cases}$ 的一拱与 $x$ 轴所围图形的面积 + + $$\begin{equation*} + A = \int_0^2\pi a y \diff x = a^2 \int_0^{2\pi} (1 - \cos t)^2 \diff t = 3\pi a^2 + \end{equation*}$$ + +### 8.1.2 极坐标 + +1. 扇形的面积：由 $\rho = \rho(\theta)$ 及射线 $\theta = \alpha$ 和 $\theta = \beta$ 所围曲边扇形的面积 + + $$\begin{equation*} + A = \int_\alpha^\beta \frac{1}{2} \rho^2(\theta) \diff \theta = \frac{1}{2} \int_\alpha^\beta \rho^2 \diff \theta + \end{equation*}$$ + +2. 阿基米德螺线：阿基米德螺线 $\rho = a \theta (a > 0)$ 对应 $\theta$ 从 $0$ 变换到 $2\pi$ 所围图形的面积 + + $$\begin{equation*} + A = \int_0^{2\pi} \frac{1}{2} (a\theta)^2\diff \theta = \frac{4}{3} \pi^3 a^2 + \end{equation*}$$ + +3. 双纽线：双纽线 $\rho^2 = a^2 \cos 2\theta$ 所围平面图形的面积 + + $$\begin{equation*} + A = 4 \int_0^{\pi/4} a^2 \cos 2\theta \diff \theta = a^2 + \end{equation*}$$ + +4. 心形线：心形线 $\rho = a(a+\cos\theta), a>0$ 所围平面图形的面积 + + $$\begin{equation*} + A = 2 \cdot \frac{1}{2} \int_0^\pi a^2 (1+ \cos \theta)^2 \diff \theta = \frac{3}{2}\pi a^2 + \end{equation*}$$ + +5. 曲线的弧长 + + | 弧微分 | 直角坐标 | 参数方程 | 极坐标 | + | :--: | :--: | :--: | :--: | + | $\diff s = \sqrt{(\diff x)^2 + (\diff y)^2}$ | $s = \displaystyle\int_a^b \sqrt{1 + y'^2}\diff x$ | $s = \displaystyle\int_a^b \sqrt{\varphi'^2(t) + \psi'^2(t)}\diff t$ | $s = \displaystyle\int_\alpha^\beta \sqrt{\rho^2{\theta} + \rho'^2(\theta)} \diff\theta$ + +## 8.2 空间 + +1. 已知平行截面面积求立体体积：令 $A(x)$ 为截面面积，则立体体积 + + $$\begin{equation*} + V = \int_a^b A(x) \diff x + \end{equation*}$$ + +2. 旋转体体积 + + | 薄片法 | 柱壳法 | + | :--: | :--: | + | $V = \displaystyle\int_a^b \pi f^2(x) \diff x$ | $V = \displaystyle\int_a^b 2\pi x \left| f(x) \right|\diff x$ |