微积分: 向量值函数常用公式

在解决实际问题时, 向量值函数的微积分运算不可避免. 本文列出了经常用到的恒等式和微分积分公式.

弱导数

  1. 对向量值函数 \(\mathbf{u}, \mathbf{v}\in L^2(\Omega)^3\), 定义内积如下$$ (\mathbf{u},\mathbf{v}) = \int_{\Omega} \sum_{i=1}^3 u_i v_i \,\mathrm{d}x .$$
  2. 分布导数. 对任意函数 \(\psi\in \mathcal{D}(\Omega)’\), 它的分布导数 \(\partial^{\alpha}\psi\) 定义为满足如下等式的唯一分布$$ (\partial^{\alpha}\psi, \phi) = (-1)^{|\alpha|} (\psi, \partial^{\alpha}\phi)\quad \forall \phi \in \mathcal{D}(\Omega).$$
  3. Sobolev 空间. 给定整数 \(k\geq 0\), 实数 \(1\leq p<\infty\), 定义$$ W^{k,p}(\Omega):=\left\{ v\in L^p(\Omega):\ \partial^\alpha v\in L^p(\Omega) \text{ for all } |\alpha|\leq k \right\}.$$

微分算子

三维空间中的梯度 (Gradient), 散度 (divergence), 旋度 (Curl):

$$\begin{align*} \nabla p &= \begin{pmatrix} \frac{\partial p}{\partial x_1} \\ \frac{\partial p}{\partial x_2} \\ \frac{\partial p}{\partial x_3} \end{pmatrix} \\ \nabla\cdot \mathbf{v} &= \frac{\partial v_1}{\partial x_1} +\frac{\partial v_2}{\partial x_2} + \frac{\partial v_3}{\partial x_3}, \\ \nabla\times \mathbf{v} &= \begin{pmatrix} \frac{\partial }{\partial x_1} \\ \frac{\partial }{\partial x_2} \\ \frac{\partial}{\partial x_3} \end{pmatrix}\times \begin{pmatrix} v_1\\ v_2 \\ v_3 \end{pmatrix} =\begin{pmatrix} \frac{\partial v_3}{\partial x_2} – \frac{\partial v_2}{\partial x_3}\\ \frac{\partial v_1}{\partial x_3} – \frac{\partial v_3}{\partial x_1}\\ \frac{\partial v_2}{\partial x_1} – \frac{\partial v_1}{\partial x_2} \end{pmatrix}\end{align*}$$

恒等式

$$ \nabla\cdot (\nabla\times \mathbf{v})=0,\quad \nabla\times (\nabla p) = 0$$

$$ \begin{align*} \mathbf{u}\times \mathbf{v} &= -\mathbf{v}\times \mathbf{u},\\ \mathbf{u}\cdot (\mathbf{v}\times \mathbf{w}) &= (\mathbf{u}\times \mathbf{v})\cdot \mathbf{w} = (\mathbf{w}\times \mathbf{u})\cdot \mathbf{w},\\ \mathbf{u}\times (\mathbf{v}\times \mathbf{w}) &= \mathbf{v}(\mathbf{u}\cdot \mathbf{w}) -\mathbf{w}(\mathbf{u}\cdot \mathbf{v}) \end{align*} $$

$$ \begin{align*} \nabla\cdot (p \mathbf{v}) &= \nabla p \cdot \mathbf{v}+p \nabla\cdot v,\\ \nabla\cdot (\mathbf{u}\times \mathbf{v}) &= (\nabla\times \mathbf{u})\cdot \mathbf{v} – \mathbf{u} \cdot (\nabla\times \mathbf{v})\\ \nabla\times (p \mathbf{v}) &= (\nabla p)\times \mathbf{v} + p \nabla\times \mathbf{v} \end{align*} $$

$$ \nabla\times (\nabla\times \mathbf{v})= \nabla (\nabla\cdot \mathbf{v})-\Delta \mathbf{v}$$

$$ \nabla\times (\mathbf{u}\times \mathbf{v}) = \mathbf{u}(\nabla\cdot \mathbf{v}) – (\mathbf{u}\cdot \nabla )\mathbf{v} + (\mathbf{v}\cdot \nabla )\mathbf{u} – \mathbf{v}(\nabla \cdot \mathbf{u})$$

如果 \(A\in \mathbb{R}^3\) 可逆且 \(\mathbf{u},\mathbf{v}\in \mathbb{R}^3\), 那么$$ (A \mathbf{u})\times (A \mathbf{v}) = \mathrm{det}(A)A^{-T}(\mathbf{u}\times \mathbf{v})$$

积分等式

定理 1 (divergence theorem) 令 \(\Omega \subset \mathbb{R}^3\) 为有界开集, 边界为 \(\partial \Omega\), 那么$$ \int_{\Omega}\nabla\cdot \mathbf{v} \,\mathrm{d}x = \int_{\partial\Omega} \mathbf{v}\cdot \mathbf{n} ds \quad\forall \mathbf{v}\in C^1(\bar{\Omega})^3.$$

如下分步积分显然成立:$$\begin{align*} \int_{\Omega} (\nabla\cdot \mathbf{v})p &= -\int_{\Omega}\mathbf{v}\cdot \nabla p + \int_{\partial\Omega}(\mathbf{n}\cdot \mathbf{v}) p ds\\ \int_{\Omega} (\Delta p) q &= -\int_{\Omega}\nabla p \cdot \nabla q+ \int_{\partial\Omega}(\mathbf{n}\cdot \nabla p) q ds \\ \int_{\Omega} (\Delta p) q – q \Delta q &= \int_{\partial\Omega}(\mathbf{n}\cdot \nabla p) q – (\mathbf{n}\cdot \nabla q) p ds \\ \int_{\Omega} (\nabla\times \mathbf{u}) \cdot \mathbf{v} &= \int_{\Omega}\mathbf{u}\cdot (\nabla\times \mathbf{v}) \,\mathrm{d}x + \int_{\partial\Omega}(\mathbf{n}\times \mathbf{u}) \cdot \mathbf{v} ds \end{align*}$$

二维空间

梯度 (Gradient), 散度 (divergence), 标量旋度 (Curl), 向量旋度 (Curl):$$\begin{align*} \nabla p &= \begin{pmatrix} \frac{\partial p}{\partial x_1} \\ \frac{\partial p}{\partial x_2} \end{pmatrix} \\ \nabla\cdot \mathbf{v} &= \frac{\partial v_1}{\partial x_1} +\frac{\partial v_2}{\partial x_2} \\ \nabla\times \mathbf{v} &= \begin{pmatrix} \frac{\partial }{\partial x_1} \\ \frac{\partial }{\partial x_2} \end{pmatrix}\times \begin{pmatrix} v_1\\ v_2 \end{pmatrix} =\begin{pmatrix} \frac{\partial v_2}{\partial x_1} – \frac{\partial v_1}{\partial x_2} \end{pmatrix} \\ \nabla\times p & =\begin{pmatrix} \frac{\partial p}{\partial x_2} \\- \frac{\partial p}{\partial x_1} \end{pmatrix}\end{align*}$$

Stokes 公式

  • 令 \(S\) 为三维空间中的曲面$$ \int_{S} \nabla\times \mathbf{u}dS = \int_{\partial\Omega}\mathbf{u}\cdot \tau ds\quad \forall \mathbf{u}\in C^1(\bar{\Omega})^3.$$
  • 二维 Stokes 公式: \(S \subset \mathbb{R}^2\), \(\forall \mathbf{u}\in C^1(\bar{S})^2\)$$ \int_{S} \nabla\times \mathbf{u} dS = \int_{\partial S}\mathbf{u}\cdot \tau ds$$

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