Go Back > New To GIMPS? Start Here! > Homework Help

Thread Tools
Old 2012-02-09, 04:22   #1

2×1,973 Posts
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.


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


(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
  Reply With Quote
Old 2012-02-09, 04:35   #2

52·89 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.
  Reply With Quote

Thread Tools

Similar Threads
Thread Thread Starter Forum Replies Last Post
Calculus davieddy Puzzles 0 2011-02-07 09:41
Review of calculus/ODE Primeinator Analysis & Analytic Number Theory 15 2011-01-12 23:05
Fractional Calculus nibble4bits Math 2 2008-01-11 21:46
Multivariable differentiability Damian Miscellaneous Math 17 2006-07-04 13:06
can somebody plz help me with this calculus? JustinJS Homework Help 4 2005-09-16 12:36

All times are UTC. The time now is 08:37.

Fri Sep 18 08:37:10 UTC 2020 up 8 days, 5:48, 0 users, load averages: 2.39, 1.97, 1.79

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2020, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.