I was wondering how one would proceed to convert between coordinate systems in $ \mathbb R^n $.
For $ \mathbb R^2 $ the conversion is easy and just basic trigonometry. Given $(r, \theta)$ we can convert these polar coordinates to cartesian coordinates to $(x=r\cos\theta, y=r\sin\theta)$
The same follows for $ \mathbb R^3 $. Given $(r,\theta_1,\theta_2)$ in spherical coordinates we can convert the spherical coordinates to cartesian coordinates with the following transformation $(x=r\cos\theta_2\cos\theta_1,y=r\cos\theta_2\sin\theta_1, z=r\sin\theta_2)$, just so my point of reference is clear:$\theta_1$is the angle lying in the x-y plane, and $\theta_2$ is the angle the vector makes with the x-y plane.
So what about for four dimensions and up? For the sake of simplicity we can define the n-sphere vector as $(r,\theta_1,\theta_2,...,\theta_n)$ and the cartesian vectors $(x_1,x_2,...,x_n)$ and maintain the same reference of $\theta_1,\theta_2$ as before.
How do we define a transformation in $ \mathbb R^4 $?
A look into this wikipedia page under the spherical coordinates section gave me something that I don't believe is quite right.
As according to the section
$(x_1=r\cos\theta_1,x_2=r\sin\theta_1 \cos\theta_2,x_3=\sin\theta_1 \sin\theta_2\cos\theta_3,...)$ which is already in contradiction to my statement about $ \mathbb R^3$ transformations.
What is the general transformation rule? Am I missing something that I should know from linear algebra that I might be forgetting because my linear algebra is rusty? If so, please let me know what I should review or point me in the right direction.