01 - MAT 473 1 1 Euclidean Space Linear Maps f(x h f(x h0 h exists When it does we call the value of the limit the derivative of f at x and denote

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MAT 473 1 1. Euclidean Space, Linear Maps. Recall that a function f : R R is diferentiable at x R if lim h 0 f ( x + h ) f ( x ) h exists. When it does, we call the value of the limit the derivative of f at x , and denote it by f ° ( x ). Equivalently, f is diFerentiable at x if there exists c R such that lim h 0 f ( x + h ) f ( x ) ch h =0 . (Take c = f ° ( x ).) Equivalently, f is diFerentiable at x if there exists a linear function T : R R such that lim h 0 f ( x + h ) f ( x ) T ( h ) h =0 . (Take T ( h )= ch .) Equivalently, f is diFerentiable at x if there exists a linear function T : R R such that lim h 0 f ( x + h ) f ( x ) T ( h ) | h | =0 . Since we can talk about limits and linear functions and 0 and we can divide by | h | in higher- dimensional spaces, this re-formulation suggests a way to de±ne derivatives of vector-valued functions of several variables. ²irst we need some notation and basic facts concerning linear maps. I’ll use the symbol R n to denote the set of all real n × 1 column vectors x = x 1 . . . x n .T o save space, column vectors are often written as n -tuples x =( x 1 ,...,x n ). (The commas distinguish an n -tuple from a 1 × n row vector.) R n is a vector space over R under the componentwise operations ( x 1 ,...,x n )+( y 1 ,...,y n )=( x 1 + y 1 ,...,x n + y n ) and c ( x 1 ,...,x n )=( cx 1 ,...,cx n ) . In fact, R n is a normed vector space with the norm de±ned by ° x ° = ° x 2 1 + ··· x 2 n , which means that (i) ° x °≥ 0 for all x ,and ° x ° = 0 if and only if x = 0 =(0 , 0 ,..., 0); (ii) ° c
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01 - MAT 473 1 1 Euclidean Space Linear Maps f(x h f(x h0 h exists When it does we call the value of the limit the derivative of f at x and denote

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