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梯度，散度与旋度是几何中重要的计算对象，形式上定义 nn 维空间上的 Nabla 算子为

∇:=(∂∂x1∂∂x2⋮∂∂xn)\nabla := \begin{pmatrix} \dfrac{\partial}{\partial x_1} \\[6pt] \dfrac{\partial}{\partial x_2} \\[6pt] \vdots \\[6pt] \dfrac{\partial}{\partial x_n} \end{pmatrix}

# 梯度

给定在开集 U⊂RnU \subset \mathbb R^n 上光滑的标量场 ff，定义其 梯度 (Gradient)「勾配」 为

grad:=∇f=(∂f∂x1∂f∂x2⋮∂f∂xn)\mathrm{grad} := \nabla f = \begin{pmatrix} \dfrac{\partial f}{\partial x_1} \\[6pt] \dfrac{\partial f}{\partial x_2} \\[6pt] \vdots \\[6pt] \dfrac{\partial f}{\partial x_n} \end{pmatrix}

梯度算子作用在标量场上，得到一个向量场

命题 梯度算子的计算性质
令 f,gf, g 为 UU 上的光滑标量场，a,b∈Ra, b \in \mathbb R，则

  1. 线性：grad(af+bg)=a gradf+b gradg\mathrm{grad}(af + bg) = a \, \mathrm{grad} f + b \, \mathrm{grad} g
  2. 乘积法则：grad(fg)=f gradg+g gradf\mathrm{grad}(fg) = f \, \mathrm{grad} g + g \, \mathrm{grad} f
证明

(1)
偏微分的线性性质给出

grad(af+bg)=(∂(af+bg)∂x1⋮∂(af+bg)∂xn)=(a∂f∂x1+b∂g∂x1⋮a∂f∂xn+b∂g∂xn)=a gradf+b gradg\mathrm{grad}(af + bg) = \begin{pmatrix} \dfrac{\partial (af + bg)}{\partial x_1} \\[6pt] \vdots \\[6pt] \dfrac{\partial (af + bg)}{\partial x_n} \end{pmatrix} = \begin{pmatrix} a \dfrac{\partial f}{\partial x_1} + b \dfrac{\partial g}{\partial x_1} \\[6pt] \vdots \\[6pt] a \dfrac{\partial f}{\partial x_n} + b \dfrac{\partial g}{\partial x_n} \end{pmatrix} = a \, \mathrm{grad} f + b \, \mathrm{grad} g

(2)
同样，继承于偏微分的乘积法则

grad(fg)=(∂(fg)∂x1⋮∂(fg)∂xn)=(f∂g∂x1+g∂f∂x1⋮f∂g∂xn+g∂f∂xn)=f gradg+g gradf\mathrm{grad}(fg) = \begin{pmatrix} \dfrac{\partial (fg)}{\partial x_1} \\[6pt] \vdots \\[6pt] \dfrac{\partial (fg)}{\partial x_n} \end{pmatrix} = \begin{pmatrix} f \dfrac{\partial g}{\partial x_1} + g \dfrac{\partial f}{\partial x_1} \\[6pt] \vdots \\[6pt] f \dfrac{\partial g}{\partial x_n} + g \dfrac{\partial f}{\partial x_n} \end{pmatrix} = f \, \mathrm{grad} g + g \, \mathrm{grad} f

□\square

几何意义上，梯度场 gradf\mathrm{grad} f 的方向是标量场 ff 增长最快的方向，且其模长表示在该方向上增长的速率

若对于一个向量场 F\boldsymbol F，存在一个标量场 ff 使得 F=gradf\boldsymbol F = \mathrm{grad} f，则称 F\boldsymbol F 为 保守场 (Conservative Field)「保存場」，此时 ff 称为 F\boldsymbol F 的 势函数 (Potential Function)「ポテンシャル関数」

# 散度

定义
令 F=(F1,F2,⋯ ,Fn)\boldsymbol F = (F_1, F_2, \cdots, F_n) 为定义在开集 U⊂RnU \subset \mathbb R^n 上的光滑向量场，则称

div F:=∇⋅F=∑i=1n∂Fi∂xi\mathrm{div} \, \boldsymbol F := \nabla \cdot \boldsymbol F = \sum_{i=1}^n \frac{\partial F_i}{\partial x_i}

为 F\boldsymbol F 的 散度 (Divergence)「発散」

# 旋度

旋度的定义本身比较复杂，但是好在实际应用中往往只需要处理二维空间中的旋度
严格意义上，输入一个向量场，输出一个同维度的向量场的旋度算子仅存在于三维空间

给定 R3\mathbb R^3 上的向量场 F=(F1,F2,F3)\boldsymbol F = (F_1, F_2, F_3)，定义其 旋度 (Curl)「回転」 为

curl F:=∇×F=(∂F3∂x2−∂F2∂x3∂F1∂x3−∂F3∂x1∂F2∂x1−∂F1∂x2)\mathrm{curl} \, \boldsymbol F := \nabla \times \boldsymbol F = \begin{pmatrix} \dfrac{\partial F_3}{\partial x_2} - \dfrac{\partial F_2}{\partial x_3} \\[6pt] \dfrac{\partial F_1}{\partial x_3} - \dfrac{\partial F_3}{\partial x_1} \\[6pt] \dfrac{\partial F_2}{\partial x_1} - \dfrac{\partial F_1}{\partial x_2} \end{pmatrix}

然而这样类似的形式在二维向量场上也被广泛使用，所以从方便的角度考虑，定义 R2\mathbb R^2 上的向量场 F=(F1,F2)\boldsymbol F = (F_1, F_2) 的 “旋度” 为

rot F:=∂F2∂x1−∂F1∂x2\mathrm{rot} \, \boldsymbol F := \frac{\partial F_2}{\partial x_1} - \frac{\partial F_1}{\partial x_2}

命题 旋度算子的计算性质
令 ff 为 R2\mathbb R^2 上的光滑标量场
令 F=(F1,F2),G=(G1,G2)\boldsymbol F = (F_1, F_2), \boldsymbol G = (G_1, G_2) 为 R2\mathbb R^2 上的光滑向量场
a,b∈Ra, b \in \mathbb R，则

  1. 线性：rot(aF+bG)=a rot F+b rot G\mathrm{rot}(a\boldsymbol F + b\boldsymbol G) = a \, \mathrm{rot} \, \boldsymbol F + b \, \mathrm{rot} \, \boldsymbol G
  2. 乘积法则：rot(fF)=f rot F+gradf×F\mathrm{rot}(f \boldsymbol F) = f \, \mathrm{rot} \, \boldsymbol F + \mathrm{grad} f \times \boldsymbol F
  3. 梯度场无旋：rot(gradf)=0\mathrm{rot}(\mathrm{grad} f) = 0
证明

(1)
偏微分算子的线性性质给出

rot(aF+bG)=∂(aF2+bG2)∂x1−∂(aF1+bG1)∂x2=a(∂F2∂x1−∂F1∂x2)+b(∂G2∂x1−∂G1∂x2)=a rot F+b rot G\begin{aligned} \mathrm{rot}(a\boldsymbol F + b\boldsymbol G) &= \frac{\partial (aF_2 + bG_2)}{\partial x_1} - \frac{\partial (aF_1 + bG_1)}{\partial x_2} \\ &= a \left(\frac{\partial F_2}{\partial x_1} - \frac{\partial F_1}{\partial x_2}\right) + b \left(\frac{\partial G_2}{\partial x_1} - \frac{\partial G_1}{\partial x_2}\right) \\ &= a \, \mathrm{rot} \, \boldsymbol F + b \, \mathrm{rot} \, \boldsymbol G \end{aligned}

(2)
利用偏微分的乘积法则

rot(fF)=∂(fF2)∂x1−∂(fF1)∂x2=f∂F2∂x1+∂f∂x1F2−(f∂F1∂x2+∂f∂x2F1)=f(∂F2∂x1−∂F1∂x2)+(∂f∂x1F2−∂f∂x2F1)=f rot F+gradf×F\begin{aligned} \mathrm{rot}(f \boldsymbol F) &= \frac{\partial (f F_2)}{\partial x_1} - \frac{\partial (f F_1)}{\partial x_2} \\ &= f \frac{\partial F_2}{\partial x_1} + \frac{\partial f}{\partial x_1} F_2 - \left(f \frac{\partial F_1}{\partial x_2} + \frac{\partial f}{\partial x_2} F_1\right) \\ &= f \left(\frac{\partial F_2}{\partial x_1} - \frac{\partial F_1}{\partial x_2}\right) + \left(\frac{\partial f}{\partial x_1} F_2 - \frac{\partial f}{\partial x_2} F_1\right) \\ &= f \, \mathrm{rot} \, \boldsymbol F + \mathrm{grad} f \times \boldsymbol F \end{aligned}

(3)
由于 ff 是光滑的，所以偏导顺序不影响结果

rot(gradf)=∂∂x1(∂f∂x2)−∂∂x2(∂f∂x1)=∂2f∂x1∂x2−∂2f∂x2∂x1=0\begin{aligned} \mathrm{rot}(\mathrm{grad} f) &= \frac{\partial}{\partial x_1}\left(\frac{\partial f}{\partial x_2}\right) - \frac{\partial}{\partial x_2}\left(\frac{\partial f}{\partial x_1}\right) \\ &= \frac{\partial^2 f}{\partial x_1 \partial x_2} - \frac{\partial^2 f}{\partial x_2 \partial x_1} = 0 \end{aligned}

□\square