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Old 2012-02-09, 04:22   #1
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Default Multivariable Calculus Problem

Let f,g:R^n -> R^n be continuously differentiable functions and suppose there is a diffeomorphism h on R^n such that:

f(x) = M(x)^{-1}g(h(x)) for any x
where M(x) is the Jacobian of h at x

Obviously, x_0 is a zero of f iff h(x_0) is a zero of g.

Show that in this case, the Jacobian of f at x_0 and the Jacobian of g at h(x_0) are similar n\times n matrices.

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When n=1, this is easy because the Jacobian is just the regular derivative and I can apply the quotient rule.

f(x)=\frac{g(h(x))}{h'(x)}
f'(x)=\frac{h'(x)g'(h(x))h'(x)-g(h(x))h''(x)}{h'(x)^2}
f'(x_0)=\frac{g'(h(x_0))h'(x_0)^2-g(h(x_0))h''(x_0)}{h'(x_0)^2}
f'(x_0)=g'(h(x_0))

(Of course, two 1\times 1 matrices are similar only when they are equal)

But how can I handle the general case? Even when n=2, the calculations get quite tedious to do by hand.

Thanks a lot
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Old 2012-02-09, 04:35   #2
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Admittedly, when dealing with the n=1 case, I took a second derivative of h, 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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