Showing that a partial derivative equation holds

[s3a]

Homework Statement

The question is attached as Question.jpg.

Homework Equations

Partial differentiation.

The Attempt at a Solution

This seems obvious to me but I don't know how to express myself mathematically. Basically, what I'd do is:

[∂(u,v)/∂(x,y)] [∂(x,y)/∂(r,s)] = [∂(u,v)/∂(r,s)] [∂(x,y)/∂(x,y)] = ∂(u,v)/∂(r,s)

as for ∂(u,v)/∂(x,y) = 1/[∂(u,v)/∂(x,y)], that's especially obvious to me! It's like saying 1/(1/2) = 1 * 2/1 = 2! Whenever I get an “It's obvious” feeling, that usually tells me that I'm not grasping the assumptions that are made when using these mathematical tools.

Any assistance in solving this problem would be greatly appreciated!

Edit: Forgot to attach the question! :P

 

[LCKurtz]

Homework Statement

The question is attached as Question.jpg.

Homework Equations

Partial differentiation.

The Attempt at a Solution

This seems obvious to me but I don't know how to express myself mathematically. Basically, what I'd do is:

[∂(u,v)/∂(x,y)] [∂(x,y)/∂(r,s)] = [∂(u,v)/∂(r,s)] [∂(x,y)/∂(x,y)] = ∂(u,v)/∂(r,s)

as for ∂(u,v)/∂(x,y) = 1/[∂(u,v)/∂(x,y)], that's especially obvious to me! It's like saying 1/(1/2) = 1 * 2/1 = 2! Whenever I get an “It's obvious” feeling, that usually tells me that I'm not grasping the assumptions that are made when using these mathematical tools.

Any assistance in solving this problem would be greatly appreciated!

Edit: Forgot to attach the question! :P

Remember that a Jacobian is the determinant of a matrix. Write out the matrix for [∂(u,v)/∂(x,y)] and [∂(x,y)/∂(r,s)]. What do you get if you multiply those matrices? And see if you can use the hint that |AB|=|A||B| for the determinant of a product of square matrices.

 

[s3a]

I attached the Jacobian stuff you asked for.

For the hint, A and B are matrices, right? What do the || mean in this case?

 

[LCKurtz]

I attached the Jacobian stuff you asked for.

For the hint, A and B are matrices, right? What do the || mean in this case?

|A| means the determinant of A. And in your last attachment, multiply the matrices themselves before calculating any determinants. I think you will find you don't have to actually evaluate any determinants to work the problem.