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lecture3 - Lecture 3 2 2.1 Dierentiation Dierentiation in n...

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Lecture 3 2 Differentiation 2.1 Differentiation in n dimensions We are setting out to generalize to n dimensions the notion of differentiation in one- dimensional calculus. We begin with a review of elementary one-dimensional calculus. Let I R be an open interval, let f : I R be a map, and let a I . Definition 2.1. The derivative of f at a is f ( a ) = lim f ( a + t ) f ( a ) , (2.1) t 0 t provided that the limit exists. If the limit exists, then f is differentiable at a . There are half a dozen or so possible reasonable generalizations of the notion of derivative to higher dimensions. One candidate generalization which you have probably already encountered is the directional derivative. Definition 2.2. Given an open set U in R n , a map f : U R m , a point a U , and a point u R n , the directional derivative of f in the direction of u at a is D u f ( a ) = lim f ( a + tu ) f ( a ) , (2.2) t 0 t provided that the limit exists. In particular, we can calculate the directional derivatives in the direction of the standard basis vectors e 1 , . . . , e n of R n , where e 1 = (1 , 0 , . . . , 0) , (2.3) e 2 = (0 , 1 , 0 , . . . , 0) , (2.4) . . . (2.5) e n = (0 , . . . , 0 , 1) . (2.6) Notation. The directional derivative in the direction of a standard basis vector e i of R n is denoted by D i f ( a ) = D e i f ( a ) = f ( a ) . (2.7) ∂x i We now try to answer the following question: What is an
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