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 April 18th, 2017, 02:13 PM #1 Member   Joined: Nov 2016 From: Kansas Posts: 65 Thanks: 0 Non-linear Equations Show that the (non-linear) equation system: $x^{2}$ + $y^{2}$ -$u^{2}$ - v=0 $x^{2}$ + $2y^{2}$ -$3u^{2}$ - $4v^{2}$=1 can be solved in (u, v) in a neighbourhood of (1/2,0,1/2,0) How to show this and also calculate the derivative of u and v wrt (x,y)?
 April 18th, 2017, 06:42 PM #2 Senior Member   Joined: Mar 2015 From: New Jersey Posts: 1,175 Thanks: 90 2xdx+2ydy=2udu+dv 2xdx+4ydy=6udu+8vdv at (1/2,0,1/2,0) dx=du dx=3du no solution
 April 18th, 2017, 07:12 PM #3 Senior Member     Joined: Sep 2015 From: Southern California, USA Posts: 1,601 Thanks: 816 I don't really know how this sort of problem is done. But if it means anything the determinant of the Jacobian is zero at that point.
April 19th, 2017, 03:35 AM   #4
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Quote:
 Originally Posted by ZMD Show that the (non-linear) equation system: $x^{2}$ + $y^{2}$ -$u^{2}$ - v=0 $x^{2}$ + $2y^{2}$ -$3u^{2}$ - $4v^{2}$=1 can be solved in (u, v) in a neighbourhood of (1/2,0,1/2,0) How to show this and also calculate the derivative of u and v wrt (x,y)?
(1/2,0,1/2,0) is not a solution of above equations. If p0 is,

to find a local solution in a neighborhood of p0, take differential of above equations and solve for dx and dy in terms of du and dv at p0.

2xdx+2ydy=2udu+dv
2xdx+4ydy=6udu+8vdv

dx= x$\displaystyle {_u}$du+x$\displaystyle {_v}$dv
dy= y$\displaystyle {_u}$du+y$\displaystyle {_v}$dv

and read off the partial derivatives.

ZMD can do the algebra.

EDIT
(1/2,0,1/2,0) is a solution of:

$x^{2}$ + $y^{2}$ -$u^{2}$ - v=0
$x^{2}$ + $2y^{2}$ +$3u^{2}$ - $4v^{2}$=1

and proceed as above.

Last edited by zylo; April 19th, 2017 at 04:11 AM.

 April 19th, 2017, 10:21 AM #5 Member   Joined: Nov 2016 From: Kansas Posts: 65 Thanks: 0 Re-edit The second equation has all +, no -
April 20th, 2017, 07:30 AM   #6
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Quote:
 Originally Posted by ZMD Re-edit The second equation has all +, no -
(1/2,0,1/2,0) is a solution either way and procedure is the same.

Last edited by skipjack; April 20th, 2017 at 11:27 AM.

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