常用 Jacobi 行列式 | 重积分变量替换

二维极坐标变换：{x=rcos⁡θy=rsin⁡θ∂(x,y)∂(r,θ)=r\left\{\begin{aligned}x=r\cos\theta\\y=r\sin\theta\end{aligned}\right.\qquad \frac{\partial(x,y)}{\partial(r,\theta)}=r{x=rcosθy=rsinθ∂(r,θ)∂(x,y)=r广义极坐标变换：{x=

陌雨’

4857人浏览 · 2021-10-16 12:42:05

陌雨’ · 2021-10-16 12:42:05 发布

二维

极坐标变换：

${x=rcos⁡θy=rsin⁡θ∂(x,y)∂(r,θ)=r\left\{\begin{aligned} x=r\cos\theta\\ y=r\sin\theta\end{aligned}\right.\qquad \frac{\partial(x,y)}{\partial(r,\theta)}=r$

广义极坐标变换：

${x=arcos⁡θy=brsin⁡θ∂(x,y)∂(r,θ)=abr\left\{\begin{aligned} x&=ar\cos\theta\\ y&=br\sin\theta\end{aligned}\right. \qquad \frac{\partial(x,y)}{\partial(r,\theta)}=abr$

三维

柱面坐标变换

${x=rcos⁡θy=rsin⁡θz=z∂(x,y,z)∂(r,θ,z)=r\left\{\begin{aligned} x&=r\cos\theta\\ y&=r\sin\theta\\ z&=z \end{aligned}\right. \qquad \frac{\partial(x,y,z)}{\partial(r,\theta,z)}=r$

球面坐标变换

${x=rsin⁡φcos⁡θy=rsin⁡φsin⁡θz=rcos⁡φ∂(x,y,z)∂(r,φ,θ)=r2sin⁡φ\left\{\begin{aligned} x&=r\sin\varphi\cos\theta\\ y&=r\sin\varphi\sin\theta\\ z&=r\cos\varphi \end{aligned}\right. \qquad \frac{\partial(x,y,z)}{\partial(r,\varphi,\theta)}=r^2\sin\varphi$

广义球面坐标变换

${x=arsin⁡φcos⁡θy=brsin⁡φsin⁡θz=crcos⁡φ∂(x,y,z)∂(r,φ,θ)=abcr2sin⁡φ\left\{\begin{aligned} x&=ar\sin\varphi\cos\theta\\ y&=br\sin\varphi\sin\theta\\ z&=cr\cos\varphi \end{aligned}\right. \qquad \frac{\partial(x,y,z)}{\partial(r,\varphi,\theta)}=abcr^2\sin\varphi$