20120209, 04:22  #1 
2^{2}·1,879 Posts 
Multivariable Calculus Problem
Let be continuously differentiable functions and suppose there is a diffeomorphism on such that:
for any where is the Jacobian of at Obviously, is a zero of iff is a zero of . Show that in this case, the Jacobian of at and the Jacobian of at are similar matrices.  When , this is easy because the Jacobian is just the regular derivative and I can apply the quotient rule. (Of course, two matrices are similar only when they are equal) But how can I handle the general case? Even when , the calculations get quite tedious to do by hand. Thanks a lot 
20120209, 04:35  #2 
5·643 Posts 
Admittedly, when dealing with the case, I took a second derivative of , despite the hypothesis only being that it is a diffeomorphism. But don't worry about that too much; this is for an applied math problem.

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