mersenneforum.org Multivariable Calculus Problem
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 2012-02-09, 04:22 #1 Unregistered   22·1,879 Posts 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. ------------- 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
 2012-02-09, 04:35 #2 Unregistered   5·643 Posts 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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