Transformations of joint probability function

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The joint pdf of X, Y, and Z is given by

$f(x,y,z) = 8xy\,\mathsf e^{-z}$ for $0≤x<y≤1, z>0$ (0 otherwise)

What is the joint pdf of $U=X/Y$ and $V = Y$?

My idea is that we should create another dummy variable for $Z$ and then use the Jacobian for 3x3 matrices. Is this correct or do I have another alternative?