Purchase Solution

Proof of Fixed Point Theorem using Stokes Theorem and Analysis

Not what you're looking for?

Ask Custom Question

Prove that if D is the closed disc |x|  1 in R2, then any map f 2 C2[D ! D] has a fixed point: f(x) = x. The proof is by contradiction, and uses Stokes theorem. Follow the steps outlined below.
(1) Define a new map F(x) = 1
Show that F has no fixed points if r is small enough.
(2) Draw the ray from F(x) to x (these are distict) and note where it cuts the circle C : |x| = 1. This point G(x) = (cos , sin ) depends smoothly on x, i.e. ...;moreover, it reduces to the identity on C.
(3) Now compute
(4) Explain why the above is a contradiction?

Purchase this Solution

Solution Summary

The solution shows a proof of a fixed point theorem using Stokes theorem and analysis. The solution is detailed and well-presented.

Purchase this Solution

Free BrainMass Quizzes
Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.

Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

Probability Quiz

Some questions on probability

Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.