Re: equation please help
From: Robert Israel (israel_at_math.ubc.ca)
Date: 11/17/04
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Date: 17 Nov 2004 08:15:01 -0500
In article <cnb0vu$2q3$1@dizzy.math.ohio-state.edu>,
Quick Function <quickcur@yahoo.com> wrote:
>What is the solution for the following equation:
>f(x, y) is a two dimensional function.
>|grad(f)| = g(x, y),
>grad is the gradient of f, g is a known function, | | is the norm.
Suppose p is an isolated zero of g. We may look for a solution
where p is a local minimum of f (of course, by symmetry f -> -f, this
could also be a local maximum; I don't know how to handle a saddle point
though). Then any point in some neighbourhood of p will lie on some
streamline of f that passes through p. For any curve C from p to q,
f(q) - f(p) = int_C grad(f).dr <= int_C g ds (where s is arc length),
with equality for the streamline. So, choosing an arbitrary value
for f(p), we define f(q) = f(p) + min_C int_C g ds, where the minimum
is taken over all rectifiable curves from p to q.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
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