Mean Value Theorem for Nonlinear Equations in R^n

[steffka]

Can someone help me... i need to show, that a system of 2 nonlinear equations
has a root. I think it is possible to use something like "mean value theorem". But i can not find any mean value theorem for R^n -> R^n.

 

[matt grime]

the mean value theorem is not true in multiple dimensions, and unless you can be more specific your problem as stated is false, there is no real solution to

x^2+y^2=1

irrespective of the second equation over R, so are you talking about C?

 

[VietDao29]

the mean value theorem is not true in multiple dimensions, and unless you can be more specific your problem as stated is false, there is no real solution to

x^2+y^2=1

Hmmm, ... Yes, it does... x = 0, y = 1 is one example.
The one does not have real root is x^2 + y^2 = -1.
Viet Dao,

 

[matt grime]

sorry, meant it to be -1 not 1.

 

[steffka]

correction resp. specification

I have two equations (nonlinear) with two variles. With some aproximative methods I get some potential roots (but not exact), so i know (or hope) there are some.
I need to show, that there exists at least one root.
I thought that somethinq like mena value theorem could help.

In attachment are equations and also the potential roots.
(the root x=0 and y=0 is trivial, but i need some other)