Mean Value Theorem for Nonlinear Equations in R^n

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Discussion Overview

The discussion revolves around the application of the Mean Value Theorem to a system of two nonlinear equations in R^n, specifically addressing the existence of roots for such equations. Participants explore the validity of the Mean Value Theorem in multiple dimensions and the conditions under which roots may exist.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant seeks assistance in demonstrating that a system of two nonlinear equations has a root, suggesting the Mean Value Theorem as a potential tool.
  • Another participant asserts that the Mean Value Theorem does not hold in multiple dimensions and questions the validity of the problem as stated, referencing the equation x^2 + y^2 = 1.
  • A different participant counters that there are indeed real solutions to the equation x^2 + y^2 = 1, providing an example of x = 0, y = 1, while noting that x^2 + y^2 = -1 has no real roots.
  • The original poster clarifies their intent, indicating they have two nonlinear equations and have identified potential roots through approximate methods, expressing a need to show the existence of at least one root beyond trivial solutions.

Areas of Agreement / Disagreement

Participants express disagreement regarding the applicability of the Mean Value Theorem in multiple dimensions and the existence of roots for the given equations. The discussion remains unresolved with competing views on these points.

Contextual Notes

Participants reference specific equations and potential roots, indicating that the discussion may depend on the definitions and assumptions related to the Mean Value Theorem and the nature of the equations involved.

steffka
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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.
 
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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?
 
matt grime said:
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,
 
sorry, meant it to be -1 not 1.
 
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)
 

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