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 Unregistered 2012-02-09 04:22

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

 Unregistered 2012-02-09 04:35

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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