I'm a little confused about this problem:

Let X and Y be independent, iid uniform random variables on [0,1]. Compute the joint density of:

a) U = X + Y, V = X/Y

b) U = X, V = X/Y

c) U = X + Y, V = X/(X+Y)

I know how to approach this problem, but I am having trouble with some of the details. I rearranged U and V in part (a) to get X=UV/(V+1) and Y=U/(V+1). I know you are supposed to integrate the function and somehow use the jacobian determinant, which I calculated to be (-X/Y^2) - (1/Y). However I am not sure how to actually calculate the joint distribution function.