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lecture6 - Lecture 6 We begin with a review of some earlier...

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Lecture 6 We begin with a review of some earlier definitions. Let δ > 0 and a R n . Euclidean ball: B δ ( a ) = { x R n : x a < δ } (2.66) Supremum ball: R δ ( a ) = { x R n : x a < δ } | | (2.67) = I 1 × · · · × I n , I j = ( a j δ, a j + δ ) . Note that the supremum ball is actually a rectangle. Clearly, B δ ( a ) R δ ( a ). We use the notation B δ = B δ (0) and R δ = R δ (0). Continuing with our review, given U open in R n , a map f : U R k , and a point a U , we defined the derivate Df ( a ) : R n R k which we associated with the matrix ∂f i Df ( a ) ∂x j ( a ) , (2.68) and we define ∂f i ∂x j ( a ) . (2.69) | Df ( a ) = sup | i,j Lastly, we define U R n to be convex if a, b U = (1 t ) a + tb U for all 0 t 1 . (2.70) Before we state and prove the Inverse Function Theorem, we give the following definition. Definition 2.13. Let U and V be open sets in R n and f : U V a C r map. The map f is is a C r diffeomorphism if it is bijective and f 1 : V U is also C r . Inverse Function Theorem. Let U be an open set in R n , f : U R n a C r map, and a U . If Df ( a ) : R n R n is bijective, then there exists a neighborhood U 1 of a in U and a neighborhood V of f ( a ) in R n such that F U 1 is a C r diffeomorphism of | U 1 at V . Proof. To prove this we need some elementary multi-variable calculus results, which we provide with the following lemmas.
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