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# lecture4 - Lecture 4 2.2 Conditions for Differentiability...

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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 1-D calculus, which we state here without proof....
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lecture4 - Lecture 4 2.2 Conditions for Differentiability...

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