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Proof of Fixed Point Theorem using Stokes Theorem and Analysis

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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?

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

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