- #1

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g(x,y) = (f(x,y), y) near an appropriate point.

2. prove #1 using the implicit function theorem.

3. generalize part 2) to show that no function

[tex] f:R^n -> R^m, [/tex] with n>m can be one-to-one.

i am not sure where to start on this problem. for #1, i don't know how to apply the hint that's given. what is this "appropriate point" it's referring to? i guess i started by trying to take the determinant of f'(x,y). but f'(x,y) is a 2x1 matrix...how do i take determinant of that? not sure where this is leading me.

not sure what to do for #2 or #3 either. my understanding of the implicit function theorem is very fuzzy. like, i understand how to use it to differentiate one variable with respect to another, but i don't know how to use it to prove the function is not one to one. any help is appreciated - thanks in advance.