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1. PRELIMINARY DISCUSSION 1.1. Goal. The goal of this assignment is to use Green's Theorem and line integrals to prove the following theorem. Theorem 1. Let S denote the closed unit ball in R2, that is, S := {x E R2 : 1-1 Assume that F : S → R2 is a function of class C2 such that F(x) = x for all x E as. Then it cannot be the case that F(x)-1 for all x E S. In an appendix we will show that Theorem 1 implies the following theorem Then for any function F : S → S of class C2, there exists some x E S such that In other words, any function F : S → S has a fixed point. Both theorems are also true if we only assume that F is continuous, rather than of class Theorem 2. Let S denote the closed unit ball in R2, ie. S (x E R2 : k 〈 1} F(x) = x C2, but the proofs are harder and we will not discuss them (although we know enough to prove them if we wished The natural generalization of Theorem 2 is true in n dimensions for every natural number n: every continuous function F from the closed n-dimensional unit ball to itself has a fixed point. This general version of the theorem is called the Brouwer Fixed Point Theorem, proved by the Dutch mathematician L.E.J Brouwer in the early 20th century 1.2. Some remarks. As you will see, if you read the Appendix, Brouwer's Fixed Point Theorem (in 2 dimensions) is deduced from Theorem 1 using a proof by contradiction. An interesting historical fact is that in his later career, Brouwer developed strong philosophical objections to this kind of mathematical argument, and as a result, he disavowed the proof of his own Fixed Point Theorem. (Brouwer's views on the philosophical foundations of mathematics were not accepted by the mathematics community. You can decide for yourself whether you feel that his proof of Theorem 2 is acceptable.) 2. THE PROBLEMS In all the problems below, S denotes the closed unit ball in the plane, i.e. S:- {x e R2 x 1}, and SS has the standard (counterclockwise) orientation. Problem 1, Assume that F = (E, F) : S → R2 is a function of class C2. Show that if as is parametrized by x = g(t) := (cost, sint) for 0

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