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Word problem involving Jacobian (Solution included)

Could someone please help me do this question (#4)? This question involves the Jacobian and while I know how to compute Jacobians, I have trouble with these kinds of problems. Basically, I don't know how to set up the problem and why it is being done the way it is. I get the impression by the question u = u(x,y,z) and v = v(x,y,z) but I don't see such a thing in the teacher's work. All I know is to use Jacobians when the problem is about a function as a function of other functions.

Any help would be GREATLY appreciated!

Thanks in advance!