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Let the variables $x$,$y$,$z$, of $f(x,y,z)$ be on the equal status \begin{align*} \int_0^1\int_0^1\int_0^1f(x,y,z)\mathrm{d}x\mathrm{d}y\mathrm{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,(3)}}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*} Now, the main question is, if there is a suitable change of variables for the following integral \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}{?}}\iiiint\limits_{\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\\ \,\\ \overset{\begin{cases} x=\,???\\ y=\,???\\ z=\,???\\ w=\,??? \end{cases}}{\overline{\overline{\hspace{4cm}}}}\quad&\,?\iiiint\limits_{V_1}f(???,???,???,???)\,????\,{\rm\,d}r\!{\rm\,d}\psi\!{\rm\,d}\theta\!{\rm\,d}\varphi\\ \end{align*} \begin{align*} =?\int_0^{\frac{\pi}{4}}\int_{???}^{???}\int_{???}^{???}\int_{0}^{???}f(???,???,???,???)\,????\,{\rm\,d}r\!{\rm\,d}\psi\!{\rm\,d}\theta\!{\rm\,d}\varphi\\ \end{align*} Wiki: N-Sphere coordinates

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  • $\begingroup$ Let $f(x,y,z)=1_{z>y>x}$, for example. then your first equation breaks down spectacularly. $\endgroup$ – user10354138 Nov 9 '18 at 6:10
  • $\begingroup$ yeah, you are right! @user10354138 Some conditions have been missed, let the variables $x$,$y$,$z$, of $f(x,y,z)$ are on the equal status. I have already added it now. $\endgroup$ – D.Matthew Feb 19 at 5:37
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Let the variables $x$,$y$,$z$,$w$, of $f(x,y,z,w)$ be on the equal status \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,(4)}}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*} We can find \begin{align*} &&&\int_0^1\int_0^1\int_0^1\int_0^1{\rm\,d}x\!{\rm\,d}y\!{\rm\,d}z\!{\rm\,d}w\\ &&=\,&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}r^3\sin^2\psi\sin\theta\,{\rm\,d}r\!{\rm\,d}\psi\!{\rm\,d}\theta\!{\rm\,d}\varphi\\ &&=\,&1 \end{align*}

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