梯度、散度、旋度
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前言
感谢视频作者，无意间刷到，把旋度和散度讲的很通俗易懂，可视化也很直观。本文为学习笔记。
结论：
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理解1：
- 散度是通量的体密度，即通量的微分；
- 旋度是环量的面密度，即环量的微分；
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理解2:
区别在于如何与\(\nabla\)进行运算：
数学概念 特点 符号 Numpy函数 梯度 数量积、哈达玛积 \(\odot\) np.multiply或者*散度 点乘、内积 \(\cdot\) np.dot旋度 叉乘、外积 \(\times\)或者 \(\otimes\) np.cross
1. 梯度（graident）
对于标量场和矢量场
1.1 对于标量场
nabla算子把数量场变成了向量场，即 \(\mathbb{R} \rightarrow \mathbb{R}^n\)：
\[ \nabla = \begin{bmatrix}\frac{\partial}{\partial x_1}, \frac{\partial}{\partial x_2}\frac{\partial}{\partial x_3}, \dots, \frac{\partial}{\partial x_n}\end{bmatrix}^T
\]
\(\nabla\)和函数的数量乘\(\nabla f\) 称为函数梯度
\[\nabla_{\boldsymbol{x}} f(\boldsymbol{x})\overset{\underset{\mathrm{def}}{}}{=}\begin{bmatrix} \frac{\partial }{\partial x_1}\\ \frac{\partial }{\partial x_2}\\ \vdots \\ \frac{\partial }{\partial x_n} \end{bmatrix} \odot f= \left[ \frac{\partial f(\boldsymbol{x})}{\partial x_1}, \frac{\partial f(\boldsymbol{x})}{\partial x_2},\cdots,\frac{\partial f(\boldsymbol{x})}{\partial x_n} \right]^T=\frac{\partial f(\boldsymbol{x})}{\partial \boldsymbol{x}}
\]
1.2 对矢量场
\(m\)维向量函数\(\boldsymbol{f}(\boldsymbol{x})=[f_1(\boldsymbol{x}),f_2(\boldsymbol{x}),\cdots,f_m(\boldsymbol{x})]\)相对于\(n\)维实向量\(x\)的梯度为\(n×m\)矩阵，\(\mathbb{R}^{m} \rightarrow \mathbb{R}^{n \times m}\) ：
\[\nabla\overset{\underset{\mathrm{def}}{}}{=}\begin{bmatrix}\frac{\partial}{\partial x_1}, \frac{\partial}{\partial x_2}\frac{\partial}{\partial x_3}, \dots, \frac{\partial}{\partial x_n}\end{bmatrix}^T
\]
\[\begin{aligned}\nabla_{\boldsymbol{x}} \boldsymbol{f}(\boldsymbol{x}) &=\frac{\partial \boldsymbol{f}(\boldsymbol{x})}{\partial \boldsymbol{x}} \\ & = \begin{bmatrix}\frac{\partial}{\partial x_1}\\ \frac{\partial}{\partial x_2}\\\frac{\partial}{\partial x_3}\\\vdots\\\frac{\partial}{\partial x_n}\end{bmatrix} \odot\boldsymbol [f_1(\boldsymbol{x}),f_2(\boldsymbol{x}),\cdots,f_m(\boldsymbol{x})] \\ &=
\begin{bmatrix}
\frac{\partial f_1(\boldsymbol{x})}{\partial x_1} &\frac{\partial f_2(\boldsymbol{x})}{\partial x_1} & \cdots & \frac{\partial f_m(\boldsymbol{x})}{\partial x_1} \\
\frac{\partial f_1(\boldsymbol{x})}{\partial x_2} &\frac{\partial f_2(\boldsymbol{x})}{\partial x_2} & \cdots & \frac{\partial f_m(\boldsymbol{x})}{\partial x_2} \\
\vdots &\vdots & \ddots & \vdots \\
\frac{\partial f_1(\boldsymbol{x})}{\partial x_n} &\frac{\partial f_2(\boldsymbol{x})}{\partial x_n} & \cdots &\frac{\partial f_m(\boldsymbol{x})}{\partial x_n} \\
\end{bmatrix} \\ \end{aligned}
\]
1.3 梯度的性质：
- 沿着梯度方向走，函数值增大
- 沿着相反于梯度的方向走，函数值减小
- 垂直于梯度方向，函数值不变
--image
2. 散度（divergence）
对于向量场
2.1 定义
\(\nabla\)和函数的点乘\(\nabla f\) 称为函数散度，某点散度代表了该点向外的通量体密度. \(\mathbb{R}^n \rightarrow \mathbb{R}\)。
设某个三维矢量场为\(f,\) 那么散度
\[\begin{aligned}\mathbf{div} (\mathbb{f}) &= \nabla \cdot f \\&=\begin{bmatrix} {\frac{\partial f_x}{\partial x} , \frac{\partial f_y}{\partial y}, \frac{\partial f_z}{\partial z} }\end{bmatrix}\begin{bmatrix} f_x \\ f_y \\f_x\end{bmatrix}\\&=\frac{\partial f_x}{\partial x} + \frac{\partial f_y}{\partial y} + \frac{\partial f_z}{\partial z}\end{aligned}
\]
这里的\([f_x, f_Y,f_z] = {f} \cdot \vec{n}\)
2.2 高斯公式
对于三维矢量，高斯公式可以表示为：
\[\iiint \limits_\Omega (\frac{\partial f_x}{\partial x} + \frac{\partial f_y}{\partial y} + \frac{\partial f_z}{\partial z}) dV = \oiint \limits_{\partial \Omega}f_zdydz+f_ydxdz+f_zdxdy
\]
--image
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物理意义：
某点散度代表了该点向外的通量体密度，其物理意义可以理解为：定量给出向量场中任一点是否为源点或汇点。若某点散度等于0，则说明其通量为0，流进=流出；若某点散度大于0，说明流出>流进，相当于一个源点(source)；若某点散度小于0，说明流出<流进，相当于一个汇点(sink)。
应用：
流体力学中不可压缩条件为：速度场的散度为0。
→ 推导:
不可压缩意味着密度为常数，根据欧拉描述下（基于场的描述）质量连续性方程：
\(\frac{d \rho}{d t}+\rho \nabla \cdot v=0\)，由于密度为常数，因此其对时间的全导数应为0，即\(\frac{d \rho}{d t}=0\)，因此速度散度\(\nabla \cdot v\)为0。
3. 旋度（curl）
3.1 定义
\(\nabla\)和函数的叉乘\(\nabla \times f\) 称为函数旋度，某点散度代表了该点局部旋涡强度(我把散度想象成动量密度，旋度想象成旋转动量密度) . \(\mathbb{R}^n \rightarrow \mathbb{R}^n\)（受限于叉乘，通常\(n\leqslant 3\)）。
\[\begin{aligned} \mathbf{curl}(f)&= \nabla \times \textbf{f}\\ &= \begin{bmatrix} {\frac{\partial f_x}{\partial x} , \frac{\partial f_y}{\partial y}, \frac{\partial f_z}{\partial z} }\end{bmatrix} \times \begin{bmatrix} f_x \\ f_y \\f_x\end{bmatrix} \\&= \begin{bmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ u & v & w\end{bmatrix} \\& = \begin{bmatrix} \frac{\partial f_z}{\partial y} - \frac{\partial f_y}{\partial z} \\ \frac{\partial f_x}{\partial z} - \frac{\partial f_z}{\partial x} \\\frac{\partial f_y}{\partial x} - \frac{\partial f_x}{\partial y} \\\end{bmatrix} \end{aligned}
\]
应用：
流体力学中：在连续介质中,速度的旋度等于角速度的两倍.
即：\(**curl(\vec v)=\nabla \times \vec{v}=2(\vec w)**\)
→ 推导: ( 没看懂😂 )
\[\begin{aligned} \nabla \times \mathbf{v}&=\nabla \times (\boldsymbol{w} \times \mathbf{r} ) \\ &=(\frac{\partial}{\partial x_i}\vec e_i) \times (w_j\vec e_j \times r_k \vec e_k) \\ &=(\frac{\partial}{\partial x_i}\vec e_i) \times (\varepsilon_{jkn}w_j r_k \vec e_n ) \\&=\frac{\partial}{\partial x_i} (w_jr_k) \varepsilon_{jkn}\varepsilon_{inm}\vec e_m \\ &=w_j\delta_{ki}\varepsilon_{jkn}\varepsilon_{inm}\vec e_m \\ &=w_j\varepsilon_{jkn}\varepsilon_{knm}\vec e_m \\ &=w_j2\delta_{jm}\vec e_m \\&=2w_j\vec e_j \end{aligned} \]
4. \(\nabla\) 、\(\Delta\)算子
\[\nabla = \begin{bmatrix}\frac{\partial}{\partial x_1}, \frac{\partial}{\partial x_2},\frac{\partial}{\partial x_3}, \dots, \frac{\partial}{\partial x_n}\end{bmatrix}^T
\]
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\(\nabla\)和\(\nabla\)做内积：
求得是 梯度的散度，即 \(\nabla \cdot \nabla\)， 为了不混淆，将它记为拉普拉斯算子，$\Delta = \nabla^2 = \nabla \cdot \nabla^T $：
\[\begin{aligned} \Delta &= \nabla^2\\ &= \nabla \cdot \nabla^T \\&= \begin{bmatrix}\frac{\partial}{\partial x_1}, \frac{\partial}{\partial x_2},\frac{\partial}{\partial x_3}, \dots, \frac{\partial}{\partial x_n}\end{bmatrix} \begin{bmatrix}\frac{\partial}{\partial x_1}\\ \frac{\partial}{\partial x_2}\\ \frac{\partial}{\partial x_3}\\ \vdots\\ \frac{\partial}{\partial x_n}\end{bmatrix} \\&= \frac{\partial^2} {\partial x_1^2} + \frac{\partial^2} {\partial x_2^2}+ \dots + \frac{\partial^2} {\partial x_n^2}\end{aligned}
\]
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\(\nabla\)和\(\nabla^T\)数量积:
求得是多元函数的二阶导，即海塞矩阵\(H=\nabla \nabla^T\)：
\[\begin{aligned}H &= \nabla \odot \nabla^T\\
&= \begin{bmatrix}\frac{\partial}{\partial x_1}\\ \frac{\partial}{\partial x_2}\\ \frac{\partial}{\partial x_3}\\ \vdots\\ \frac{\partial}{\partial x_n}\end{bmatrix} \odot \begin{bmatrix}\frac{\partial}{\partial x_1}, \frac{\partial}{\partial x_2},\frac{\partial}{\partial x_3}, \dots, \frac{\partial}{\partial x_n}\end{bmatrix} \\
&= \begin{bmatrix} \frac{\partial^2} {\partial x_1^2} & \cdots &\frac{\partial^2} {{\partial x_1} {\partial x_n}} \\ {\vdots}&{\ddots} & {\vdots}\\ \frac{\partial^2} {{\partial x_1} {\partial x_n}} & \cdots & \frac{\partial^2} {\partial x_n^2} \end{bmatrix} \end{aligned}
\]
