Finding probability distribution of quantity depending on other distributions
I have a vector that depends on the coordinates of randomly drawn unit vectors in :
Here, , are drawn uniformly random from the unit circle, and are thus angles that parametrize unit vectors in . I am interested in figuring out a probability distribution for the coordinates of the -vector, but I am not well-versed in probability theory. I have been told that it is possible to somehow find a probability distribution for a quantity that depends on other distributions by somehow using the Jacobian matrix, but I have been unable to figure out how on my own.
My questions are thus:
Is there a method for finding a probability distribution for a quantity that depends on other distributions? If so, how? Does it have a name so I can learn about form other sources?
If the method exists, how would I use it on my concrete example?