This is my first proof and is most likely going to be crude. Please while reading comment tips about ways to improve my writing.
Polar coordinates are transformed through: $$x=r\cos(\theta) \qquad y=r\sin(\theta).$$ Spherical coordinates are transformed through: $$x=r\sin(\phi)\cos(\theta) \qquad y=r\sin(\phi)\sin(\theta) \qquad z=r\cos(\phi).$$ 4th dimensional spherical coordinates, or hyperspherical coordinates, are transformed through: $$x=r\sin(\epsilon)\sin(\phi)\cos(\theta) \qquad y=r\sin(\epsilon)\sin(\phi)\sin(\theta) \qquad z=r\sin(\epsilon)\cos(\phi) \qquad w=r\cos(\epsilon).$$
For 2 dimensions, the Jacobian is: $$\frac{\partial(x,y)}{\partial(r,\theta)}=\begin{bmatrix}\cos(\theta)&-r\sin(\theta)\\\sin(\theta)&r\cos(\theta)\end{bmatrix}=r.$$
For 3 Dimensions, the Jacobian is: $$\frac{\partial(x,y,z)}{\partial(r,\theta,\phi)}=\begin{bmatrix}\sin(\phi)\cos(\theta)&-r\sin(\phi)\sin(\theta)&r\cos(\phi)\cos(\theta)\\\sin(\phi)\sin(\theta)&r\sin(\phi)\cos(\theta)&r\cos(\phi)\sin(\theta)\\\cos(\phi)&0&-r\sin(\phi)\end{bmatrix}=-r^2\sin(\phi).$$
For 4 Dimensions, the Jacobinan is: $$\frac{\partial(x,y,z,w)}{\partial(r,\theta,\phi,\epsilon)}=\begin{bmatrix}\sin(\epsilon)\sin(\phi)\cos(\theta)&-r\sin(\epsilon)\sin(\phi)\sin(\theta)&r\sin(\epsilon)\cos(\phi)\cos(\theta)&r\cos(\epsilon)\sin(\phi)\cos(\theta)\\\sin(\epsilon)\sin(\phi)\sin(\theta)&r\sin(\epsilon)\sin(\phi)\cos(\theta)&r\sin(\epsilon)\cos(\phi)\sin(\theta)&r\cos(\epsilon)\sin(\phi)\sin(\theta)\\\sin(\epsilon)\cos(\phi)&0&-r\sin(\epsilon)\sin(\phi)&r\cos(\epsilon)\cos(\phi)\\\cos(\epsilon)&0&0&-r\sin(\epsilon)\end{bmatrix}\\=r^3\sin^2(\epsilon)\sin(\phi).$$
The pattern starts to arise when raising the dimension: raising the power and adding sines. After changing unique angles to $\theta_k$, the Jacobian for $n$-dimensional polar coordinates, defined by $J_n$ is: $$J_n=(-1)^{n}r^{n-1}\prod_{k=2}^{n}\sin^{k-2}(\theta_k).$$ Alternatively, $J_n$ is defined through: $$\frac{\partial(x_1,x_2,x_3,\ldots,x_n)}{\partial(r,\theta_2,\theta_3,\ldots,\theta_k)}=\begin{bmatrix}\dfrac{\partial{x_1}}{\partial{r}}&\dfrac{\partial{x_1}}{\partial{\theta_2}}&\dfrac{\partial{x_1}}{\partial{\theta_3}}&\ldots&\dfrac{\partial{x_1}}{\partial{\theta_k}}\\\dfrac{\partial{x_2}}{\partial{r}}&\dfrac{\partial{x_2}}{\partial{\theta_2}}\\\dfrac{\partial{x_3}}{\partial{r}}&&\ddots&&\vdots\\\vdots\\\dfrac{\partial{x_k}}{\partial{r}}&&\ldots&&\dfrac{\partial{x_k}}{\partial{\theta_k}}\end{bmatrix}.$$
In integration with multiple integrals, the differential can be found through a specific formula. $$\mathrm dH_{Vn}=\left|{\frac{\partial(x_1,x_2,x_3,\ldots,x_n)}{\partial(r,\theta_2,\theta_3,\ldots,\theta_k)}}\right|\,\mathrm d\bar{H}_{Vn},$$ where $H_{Vn}$ is the hypervolume of dimension $n$ and $\mathrm d\bar{H}_{Vn}=\mathrm dr\,\mathrm d\theta_2\,\mathrm d\theta_3\ldots\mathrm d\theta_k$.
The formula can be simplified by defining $C_n$ as: $|J_n|$, transforming into: $$\mathrm dH_{Vn}=C_{n}\,\mathrm d\bar{H}_{Vn}.$$ So, in integration with $n$ integrals, the polar transformation is: $$\int\cdots\int{f(x_1,x_2,\ldots,x_n)}\,\mathrm dx_1\,\mathrm dx_2\ldots\mathrm dx_n \rightarrow\int\cdots\int{f(g_1(r,\theta_2,\theta_3,\ldots,\theta_n),g_2(r,\theta_2,\theta_3,\ldots,\theta_n),\ldots,g_n(r,\theta_2,\theta_3,\ldots,\theta_n))}C_{n}\,\mathrm d\bar{H}_{Vn},$$ which completes the general formula for the Jacobian and Integration of $n$-dimensional polar coordinates.
