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在圆柱和球坐标系中的del

Wikipedia · 2025-12-26 维基百科 👁 1 阅读

下面是常用于正交中的一些向量微积分公式。

注释

  • 本文对球坐标使用标准符号ISO 80000-2，它取代了ISO 31-11，（部分其他来源可能有着颠倒θ和φ的定义）：

**极角表示为θ：它是在z轴与连接原点和目标点的径向向量之间的角度。
**方位角表示为φ：它是在x轴与径向向量在xy面上的投影之间的角度。

  • 函数可以用于替代数学函数。这是由于它的定义域和像的缘故，经典arctan函数的像为，而atan2定义的像为。

坐标转换
单位向量转换
:本页对极角采用\theta对方位角采用\varphi，这是在物理学中常用的符号。某些来源在这些公式中对方位角采用\theta对极角采用\varphi，这是常用数学符号，如果需要这种数学公式，可对换上表公式中的\theta和\varphi。

非平凡的演算规则

\operatorname{div} \, \operatorname{grad} f \equiv \nabla \cdot \nabla f \equiv \nabla^2 f

\operatorname{curl} \, \operatorname{grad} f \equiv \nabla \times \nabla f = \mathbf 0

\operatorname{div} \, \operatorname{curl} \mathbf{A} \equiv \nabla \cdot (\nabla \times \mathbf{A}) = 0

\operatorname{curl} \, \operatorname{curl} \mathbf{A} \equiv \nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}（del的拉格朗日公式）

\nabla^2 (f g) = f \nabla^2 g + 2 \nabla f \cdot \nabla g + g \nabla^2 f

直角坐标系推导
\begin{align}\operatorname{div} \mathbf A = \lim_{V\to 0} \frac{\iint_{\partial V} \mathbf A \cdot d\mathbf{S}}{\iiint_V dV}

&= \frac{A_x(x+dx)dydz - A_x(x)dydz + A_y(y+dy)dxdz - A_y(y)dxdz + A_z(z+dz)dxdy - A_z(z)dxdy}{dxdydz} \\

&= \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z}\end{align}

\begin{align}(\operatorname{curl} \mathbf A)_x = \lim_{S^{\perp \mathbf{\hat x}}\to 0} \frac{\int_{\partial S} \mathbf A \cdot d\mathbf{\ell}}{\iint_{S} dS}
&= \frac{A_z(y+dy)dz - A_z(y)dz + A_y(z)dy - A_y(z+dz)dy }{dydz} \\
&= \frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z}\end{align}

(\operatorname{curl} \mathbf A)_y和(\operatorname{curl} \mathbf A)_z的表达式可以同理得出。
註：第一式中的A_x(x+dx)是A_x在x+dx時的量值，並非A_x值乘上x+dx。以下圓柱座標、球座標的推導中亦然。

圆柱坐标系推导
:\begin{align}
\operatorname{div} \mathbf A &= \lim_{V\to 0} \frac{\iint_{\partial V} \mathbf A \cdot d\mathbf{S}}{\iiint_V dV} \\
&= \frac{A_\rho(\rho+d\rho)(\rho+d\rho)d\phi dz - A_\rho(\rho)\rho d\phi dz + A_\phi(\phi+d\phi)d\rho dz - A_\phi(\phi)d\rho dz + A_z(z+dz)d\rho (\rho +d\rho/2)d\phi - A_z(z)d\rho (\rho +d\rho/2) d\phi}{\rho d\phi d\rho dz} \\
&= \frac 1 \rho \frac{\partial (\rho A_\rho)}{\partial \rho} + \frac 1 \rho \frac{\partial A_\phi}{\partial \phi} + \frac{\partial A_z}{\partial z}
\end{align}

:\begin{align}
(\operatorname{curl} \mathbf A)_\rho &= \lim_{S^{\perp \boldsymbol{\hat \rho}}\to 0} \frac{\int_{\partial S} \mathbf A \cdot d\mathbf{\ell}}{\iint_{S} dS} \\
&= \frac{A_\phi (z)(\rho+d\rho)d\phi - A_\phi(z+dz)(\rho+d\rho)d\phi + A_z(\phi + d\phi)dz - A_z(\phi)dz}{(\rho+d\rho)d\phi dz} \\
&= -\frac{\partial A_\phi}{\partial z} + \frac{1}{\rho} \frac{\partial A_z}{\partial \phi}
\end{align}

:\begin{align}
(\operatorname{curl} \mathbf A)_\phi &= \lim_{S^{\perp \boldsymbol{\hat \phi}}\to 0} \frac{\int_{\partial S} \mathbf A \cdot d\mathbf{\ell}}{\iint_{S} dS} \\
&= \frac{A_z (\rho)dz - A_z(\rho + d\rho)dz + A_\rho(z+dz)d\rho - A_\rho(z)d\rho}{d\rho dz} \\
&= -\frac{\partial A_z}{\partial \rho} + \frac{\partial A_\rho}{\partial z}
\end{align}

:\begin{align}
(\operatorname{curl} \mathbf A)_z &= \lim_{S^{\perp \boldsymbol{\hat z}}\to 0} \frac{\int_{\partial S} \mathbf A \cdot d\mathbf{\ell}}{\iint_{S} dS} \\
&= \frac{A_\rho(\phi)d\rho - A_\rho(\phi + d\phi)d\rho + A_\phi(\rho + d\rho)(\rho + d\rho)d\phi - A_\phi(\rho)\rho d\phi}{\rho d\rho d\phi} \\
&= -\frac{1}{\rho}\frac{\partial A_\rho}{\partial \phi} + \frac{1}{\rho} \frac{\partial (\rho A_\phi)}{\partial \rho}
\end{align}

:\begin{align}
\operatorname{curl} \mathbf A &= (\operatorname{curl} \mathbf A)_\rho \hat{\boldsymbol \rho} + (\operatorname{curl} \mathbf A)_\phi \hat{\boldsymbol \phi} + (\operatorname{curl} \mathbf A)_z \hat{\boldsymbol z} \\
&= \left(\frac{1}{\rho} \frac{\partial A_z}{\partial \phi} -\frac{\partial A_\phi}{\partial z} \right) \hat{\boldsymbol \rho} + \left(\frac{\partial A_\rho}{\partial z}-\frac{\partial A_z}{\partial \rho} \right) \hat{\boldsymbol \phi} + \frac{1}{\rho}\left(\frac{\partial (\rho A_\phi)}{\partial \rho} - \frac{\partial A_\rho}{\partial \phi} \right) \hat{\boldsymbol z}
\end{align}

球坐标系推导
\begin{align}\operatorname{div} \mathbf A &= \lim_{V\to 0} \frac{\iint_{\partial V} \mathbf A \cdot d\mathbf{S}}{\iiint_V dV} \\
&= \frac{A_r(r+dr)(r+dr)d\theta\, (r+dr)\sin\theta d\phi - A_r(r)rd\theta\, r\sin\theta d\phi + A_\theta(\theta+d\theta)\sin(\theta + d\theta)\,r dr d\phi - A_\theta(\theta)\sin(\theta)\,r dr d\phi + A_\phi(\phi + d\phi) (r + dr/2)dr d\theta - A_\phi(\phi)(r + dr/2)dr d\theta}{dr\,rd\theta\,r\sin\theta d\phi} \\
&= \frac{1}{r^2}\frac{\partial (r^2A_r)}{\partial r} + \frac{1}{r \sin\theta} \frac{\partial(A_\theta\sin\theta)}{\partial \theta} + \frac{1}{r \sin\theta} \frac{\partial A_\phi}{\partial \phi}
\end{align}

\begin{align}(\operatorname{curl} \mathbf A)_r = \lim_{S^{\perp \boldsymbol{\hat r}}\to 0} \frac{\int_{\partial S} \mathbf A \cdot d\mathbf{\ell}}{\iint_{S} dS}
&= \frac{A_\theta(\phi)\,r d\theta + A_\phi(\theta + d\theta)\,r \sin(\theta + d\theta) d\phi

  • A_\theta(\phi + d\phi)\,r d\theta - A_\phi(\theta)\,r\sin(\theta) d\phi}{r d\theta\,r\sin\theta d\phi} \\

&= \frac{1}{r\sin\theta}\frac{\partial(A_\phi \sin\theta)}{\partial \theta}

  • \frac{1}{r\sin\theta} \frac{\partial A_\theta}{\partial \phi}\end{align}

\begin{align}(\operatorname{curl} \mathbf A)_\theta = \lim_{S^{\perp \boldsymbol{\hat \theta}}\to 0} \frac{\int_{\partial S} \mathbf A \cdot d\mathbf{\ell}}{\iint_{S} dS}
&= \frac{A_\phi(r)\,r \sin\theta d\phi + A_r(\phi + d\phi)dr

  • A_\phi(r+dr)(r+dr)\sin\theta d\phi - A_r(\phi)dr}{dr \, r \sin \theta d\phi} \\

&= \frac{1}{r\sin\theta}\frac{\partial A_r}{\partial \phi}

  • \frac{1}{r} \frac{\partial (rA_\phi)}{\partial r}\end{align}

\begin{align}(\operatorname{curl} \mathbf A)_\phi = \lim_{S^{\perp \boldsymbol{\hat \phi}}\to 0} \frac{\int_{\partial S} \mathbf A \cdot d\mathbf{\ell}}{\iint_{S} dS}
&= \frac{A_r(\theta)dr + A_\theta(r+dr)(r+dr)d\theta

  • A_r(\theta+d\theta)dr - A_\theta(r)\, r d\theta}{(r+dr/2) dr d\theta} \\

&= \frac{1}{r}\frac{\partial(rA_\theta)}{\partial r}

  • \frac{1}{r} \frac{\partial A_r}{\partial \theta}\end{align}

\operatorname{curl} \mathbf A = (\operatorname{curl} \mathbf A)_r \, \hat{\boldsymbol r} + (\operatorname{curl} \mathbf A)_\theta \, \hat{\boldsymbol \theta} + (\operatorname{curl} \mathbf A)_\phi \, \hat{\boldsymbol \phi} = \frac{1}{r\sin\theta} \left(\frac{\partial(A_\phi \sin\theta)}{\partial \theta}-\frac{\partial A_\theta}{\partial \phi} \right) \hat{\boldsymbol r} +\frac{1}{r} \left(\frac{1}{\sin\theta}\frac{\partial A_r}{\partial \phi} - \frac{\partial (rA_\phi)}{\partial r} \right) \hat{\boldsymbol \theta} + \frac{1}{r}\left(\frac{\partial(rA_\theta)}{\partial r} - \frac{\partial A_r}{\partial \theta} \right) \hat{\boldsymbol \phi}

单位向量转换公式
坐标参数u的单位向量以如下方式定义，u的小的正值改变导致位置向量\boldsymbol\vec{r}在\boldsymbol\hat{u}方向上的改变。因此：

:{\partial\boldsymbol\vec{r} \over \partial u}={\partial{s} \over \partial u}{\boldsymbol\hat{u}}

这里的s是弧长参数。

对于两组坐标系u_i和v_j，依据链式法则：

:d\boldsymbol\vec{r}=\sum_{i}{\partial{\boldsymbol\vec{r}}\over\partial u_i}du_i=\sum_{i}{\partial{s}\over\partial u_i}\boldsymbol\hat{u_i}du_i=\sum_{j}{\partial{s}\over\partial v_j}\boldsymbol\hat{v_j}dv_j=\sum_{j}{\partial{s}\over\partial v_j}\boldsymbol\hat{v_j}\sum_{i}{\partial{v_j}\over\partial u_i}du_i=\sum_{i}\sum_{j}{\partial{s}\over\partial v_j}{\partial{v_j}\over\partial u_i}\boldsymbol\hat{v_j}du_i

现在，使除了一个之外的所有du_i=0并在两边除以对应的坐标参数的微分，得到：
:{\partial{s}\over\partial u_i}\boldsymbol\hat{u_i}=\sum_{j}{\partial{s}\over\partial v_j}{\partial{v_j}\over\partial u_i}\boldsymbol\hat{v_j}

参见

  • Del算子
  • 正交坐标系
  • 曲线坐标系
  • 在圆柱和球坐标中的向量场

引用
外部链接

#坐标系 #向量分析

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