This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 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 multivariable calculus results, which we provide with the following lemmas....
View
Full
Document
This note was uploaded on 10/02/2010 for the course MAT unknown taught by Professor Unknown during the Fall '04 term at MIT.
 Fall '04
 unknown

Click to edit the document details