My question is very simple, I have a probability density function that depends on four variables p(x1,x2,x3,x4). Now each of those variables depend on three new variables, i.e xi=xi(y1,y2,y3) so I want to find the new PDF given by p(y1,y2,y3). The problem is that the jacobian of this transformation does not exist because the jacobian matrix is not square. So how can I find the probability of (y1+dy1,y2+dy2,y3+dy3) so that when I integrate p in y1,y2 and y3 the result is 1??

Any help or suggestion will be highly appreciated.

Thank you