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

直角坐标: \(\mathbf{e}_1=(1,0,0),\mathbf{e}_2=(0,1,0),\mathbf{e}_3=(0,0,1)\)\[ \mathbf{x}=\begin{bmatrix} x_1, x_2, x_3 \end{bmatrix} = x_1 \mathbf{e}_1 + x_2 \mathbf{e}_2 + x_3 \mathbf{e}_3\]

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

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

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

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

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

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

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: \(S \subset \mathbb{R}^2\), \(\forall \mathbf{u}\in C^1(\bar{S})^2\)$$ \int_{S} \nabla\times \mathbf{u} dS = \int_{\partial S}\mathbf{u}\cdot \tau ds$$

球坐标系

单位向量\[\begin{align*} \mathbf{e}_r &= \sin \theta \cos \phi \mathbf{e}_1 +\sin \theta \sin \phi \mathbf{e}_2 + \cos \theta \mathbf{e}_3\\ \mathbf{e}_\theta &= \cos \theta \cos \phi \mathbf{e}_1 +\cos \theta \sin \phi \mathbf{e}_2 – \sin \theta \mathbf{e}_3\\ \mathbf{e}_\phi &= -\sin \phi \mathbf{e}_1 +\cos \phi \mathbf{e}_2\end{align*}\] \[ r=\sqrt{x^2+y^2+z^2}, \theta=\arccos \left( \frac{z}{r} \right), \phi = \arctan \left( \frac{y}{x} \right)\] \[ \mathbf{x}= r \sin \theta \cos \phi \mathbf{e}_1 + r \sin \theta \sin \phi \mathbf{e}_2 + r \cos \theta \mathbf{e}_3\] \[ J^T = \begin{bmatrix} \sin \theta \cos \phi & \sin \theta \sin \phi & \cos \theta \\ r\cos \theta \cos \phi & r \cos \theta \sin \phi & -r\sin \theta \\ -r\sin \theta \sin \phi & r\sin \theta \cos\phi & 0 \\ \end{bmatrix}\] \[ J^{-T}=\begin{bmatrix}\sin\theta \cos \phi & \frac{\cos \theta\cos \phi}{r} & – \frac{\sin \phi}{r \sin \theta}\\ \sin \theta\sin \phi & \frac{\cos \theta\sin \phi}{r} & \frac{\cos{\left(\phi \right)}}{r \sin{\left(\theta \right)}}\\ \cos \theta & – \frac{\sin \theta}{r} & 0\end{bmatrix}\]

算子表达式\[ \nabla f = J^{-T}\nabla_{r,\theta,\phi}f =\frac{\partial f}{\partial r} \mathbf{e}_r + \frac{1}{r}\frac{\partial f}{\partial \theta} \mathbf{e}_\theta + \frac{1}{r \sin \theta}\frac{\partial f}{\partial \phi} \mathbf{e}_\phi\]\[ \nabla\cdot \mathbf{v} = \frac{1}{r^2} \frac{\partial }{\partial r} (r^2 v_{r}) + \frac{1}{r \sin\theta} \frac{\partial }{\partial \theta} (\sin (\theta)v_\theta) + \frac{1}{r \sin\theta}\frac{\partial v_{\phi}}{\partial \phi}\]\[ \begin{align*} \nabla\times \mathbf{v} =& \frac{1}{r \sin \theta} \left( \frac{\partial }{\partial \theta} (\sin \theta v_{\phi})-\frac{\partial v_{\theta}}{\partial \phi} \right) \mathbf{e}_{r} \\ & + \frac{1}{r} \left( \frac{1}{\sin\theta}\frac{\partial v_r}{\partial \phi}-\frac{\partial}{\partial r}(rv_{\phi}) \right) \mathbf{e}_{\theta} \\ & + \frac{1}{r} \left( \frac{\partial }{\partial r}(rv_{\theta})-\frac{\partial v_r}{\partial \theta} \right) \mathbf{e}_{\phi} \\ \end{align*}\]\[ \Delta f = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial f}{\partial r} \right) + \frac{1}{r^2 \sin\theta} \frac{\partial }{\partial \theta} \left( \sin \theta \frac{\partial f}{\partial \theta} \right) + \frac{1}{r^2 \sin^2\theta}\frac{\partial ^2 f}{\partial \phi^2}\]

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