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有关重积分的四维空间(超)球坐标变换

本帖最后由 青青子衿 于 2018-11-13 12:59 编辑

\begin{align*}
\int_0^1\int_0^1\int_0^1f(x,y,z){\rm\,d}x\!{\rm\,d}y\!{\rm\,d}z
&=\,\iiint\limits_{\substack{0\,\leqslant\,x\,\leqslant\,1 \\ 0\,\leqslant\,y\,\leqslant\,1\\ 0\,\leqslant\,z\,\leqslant\,1}}f(x,y,z){\rm\,d}x\!{\rm\,d}y\!{\rm\,d}z\\
&=\,{\color{red}{3}}\iiint\limits_{\,V_{1,(3)}\colon\substack{0\,\leqslant\,y\,\leqslant\,x\,\leqslant\,1 \\ 0\,\leqslant\,z\,\leqslant\,x\,\leqslant\,1}}f(x,y,z){\rm\,d}x\!{\rm\,d}y\!{\rm\,d}z\\
\,\\
\overset{\begin{cases}
x=r\sin\theta\cos\varphi\\
y=r\sin\theta\sin\varphi\\
z=r\cos\theta\\
\end{cases}}{\overline{\overline{\hspace{4cm}}}}\quad&\,3\iiint\limits_{V_1}f(r\sin\theta\cos\varphi,r\sin\theta\sin\varphi,r\cos\theta)\,r^2\sin\theta\,{\rm\,d}r\!{\rm\,d}\theta\!{\rm\,d}\varphi\\
\end{align*}
\begin{align*}
=\boxed{3\int_0^{\frac{\pi}{4}}\int_{\arctan\left(\sec\varphi\right)}^{\frac{\pi}{2}}\int_{0}^{\frac{1}{\sin\theta\cos\varphi}}f(r\sin\theta\cos\varphi,r\sin\theta\sin\varphi,r\cos\theta)\,r^2\sin\theta\,{\rm\,d}r\!{\rm\,d}\theta\!{\rm\,d}\varphi}\\
\end{align*}
\begin{align*}
\int_0^1\int_0^1\int_0^1\int_0^1f(x,y,z,w){\rm\,d}x\!{\rm\,d}y\!{\rm\,d}z\!{\rm\,d}w  
&=\,\iiiint\limits_{\substack{0\,\leqslant\,x\,\leqslant\,1 \\ 0\,\leqslant\,y\,\leqslant\,1\\ 0\,\leqslant\,z\,\leqslant\,1\\0\,\leqslant\,w\,\leqslant\,1}}f(x,y,z,w){\rm\,d}x\!{\rm\,d}y\!{\rm\,d}z\!{\rm\,d}w\\  
&=\,{\color{red}{4}}\iiiint\limits_{\,V_{1,(4)}\colon\substack{0\,\leqslant\,y\,\leqslant\,x\,\leqslant\,1 \\  
0\,\leqslant\,z\,\leqslant\,x\,\leqslant\,1\\
\\ 0\,\leqslant\,w\,\leqslant\,x\,\leqslant\,1}}f(x,y,z,w){\rm\,d}x\!{\rm\,d}y\!{\rm\,d}z\!{\rm\,d}w\\  
\,\\  
\overset{\begin{cases}  
x=r\sin\psi\sin\theta\cos\varphi\\  
y=r\sin\psi\sin\theta\sin\varphi\\  
z=r\sin\psi\cos\theta\\  
w=r\cos\psi
\end{cases}}{\overline{\overline{\hspace{4cm}}}}\quad&\,4\iiiint\limits_{V_1}f(\cdots,\cdots,\cdots,\cdots)\,r^3\sin^2\psi\sin\theta\,{\rm\,d}r\!{\rm\,d}\psi\!{\rm\,d}\theta\!{\rm\,d}\varphi\\  
\end{align*}
\begin{align*}
=\boxed{4\int_0^{\frac{\pi}{4}}\int_{\arctan\left(\sec\varphi\right)}^{\frac{\pi}{2}}\int_{\arctan\left(\csc\theta\sec\varphi\right)}^{\frac{\pi}{2}}\int_{0}^{\csc\psi\csc\theta\sec\varphi}f(\cdots,\cdots,\cdots,\cdots)\,r^3\sin^2\psi\sin\theta\,{\rm\,d}r\!{\rm\,d}\psi\!{\rm\,d}\theta\!{\rm\,d}\varphi}\\
\end{align*}
...
  1. 4\int_0^{\frac{\pi}{4}}\int_{\arctan\left(\sec\varphi\right)}^{\frac{\pi}{2}}\int_{\arctan\left(\sec\left(\varphi\right)\csc\left(\theta\right)\right)}^{\frac{\pi}{2}}\int_0^{\frac{1}{\sin\left(\psi\right)\sin\left(\theta\right)\cos\left(\varphi\right)}}r^3\left(\sin\psi\right)^2\sin\left(\theta\right)drd\psi d\theta d\varphi
复制代码
...
  1. 4\int_0^{\frac{\pi}{4}}\int_{\arctan\left(\sec\varphi\right)}^{\frac{\pi}{2}}\int_{\arctan\left(\csc\left(\theta\right)\sec\left(\varphi\right)\right)}^{\frac{\pi}{2}}\int_0^{\frac{1}{\sin\left(\psi\right)\sin\left(\theta\right)\cos\left(\varphi\right)}}r^4\left(\sin\psi\right)^2\sin\left(\theta\right)drd\psi d\theta d\varphi
  2. \frac{4}{5}\int_0^{\frac{\pi}{4}}\int_{\arctan\left(\sec\varphi\right)}^{\frac{\pi}{2}}\int_{\arctan\left(\csc\left(\theta\right)\sec\left(\varphi\right)\right)}^{\frac{\pi}{2}}\frac{1}{\left(\sin\psi\right)^3\left(\sin\theta\right)^4\left(\cos\varphi\right)^5}d\psi d\theta d\varphi
  3. \frac{4}{5}\int_0^{\frac{\pi}{4}}\int_{\arctan\left(\sec\varphi\right)}^{\frac{\pi}{2}}\frac{\int_0^{\sin\left(\theta\right)\cos\left(\varphi\right)}\sqrt{1+u^2}du}{\left(\sin\theta\right)^4\left(\cos\varphi\right)^5}d\theta d\varphi
  4. \frac{4}{10}\int_0^{\frac{\pi}{4}}\int_{\arctan\left(\sec\varphi\right)}^{\frac{\pi}{2}}\left(\frac{\sqrt{1+\left(\sin\left(\theta\right)\cos\left(\varphi\right)\right)^2}}{\left(\sin\theta\right)^3\left(\cos\varphi\right)^4}+\frac{\ln\left(\sin\left(\theta\right)\cos\left(\varphi\right)+\sqrt{1+\left(\sin\left(\theta\right)\cos\left(\varphi\right)\right)^2}\right)}{\left(\sin\theta\right)^4\left(\cos\varphi\right)^5}\right)d\theta d\varphi
复制代码
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