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Unregistered 2012-02-09 04:22

Multivariable Calculus Problem
 
Let [tex]f,g:R^n -> R^n[/tex] be continuously differentiable functions and suppose there is a diffeomorphism [tex]h[/tex] on [tex]R^n[/tex] such that:

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

Obviously, [tex]x_0[/tex] is a zero of [tex]f[/tex] iff [tex]h(x_0)[/tex] is a zero of [tex]g[/tex].

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

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

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

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

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

Thanks a lot

Unregistered 2012-02-09 04:35

Admittedly, when dealing with the [tex]n=1[/tex] case, I took a second derivative of [tex]h[/tex], 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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