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 4 2.2 Conditions for Differentiability In this lecture we will discuss conditions that guarantee differentiability. First, we begin with a review of important results from last lecture. Let U be an open subset of R n , let f : U R n be a map, and let a U . We defined f to be differentiable at a if there exists a linear map B : R n R m such that for h R n { } , f ( a + h ) f ( a ) Bh 0 as h . (2.24)  h  If such a B exists, then it is unique and B = Df ( a ). The matrix representing B is f i the Jacobian matrix J f ( a ) = x j ( a ) , where f = ( f 1 , . . . , f m ). Note that the mere existence of all of the partial derivatives in the Jacobian matrix does not guarantee differentiability. Now we discuss conditions that guarantee differentiability. f i Theorem 2.7. Suppose that all of the partial derivatives x j in the Jacobian matrix exist at all points x U , and that all of the partial derivatives are continuous at x = a . Then f is differentiable at a . Sketch of Proof. This theorem is very elegantly proved in Munkres, so we will simply give the general ideas behind the proof here. First, we look at the case n = 2 , m = 1. The main ingredient in the proof is the Mean Value Theorem from 1D calculus, which we state here without proof....
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