lecture3

an the point la can be written as the sum j la aj

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Unformatted text preview: tch of Proof. Note that for h �= 0 in Rn , f (a + h) − f (a) − B h → 0 as h → 0 |h| (2.19) f (a + h) − f (a) − B h → 0 as h → 0. (2.20) implies that From this you can conclude that f is continuous at a. Remark. Let L : Rn → Rm be a linear map and a ∈ Rn . The point a can be written � as a sum a = n=1 aj ej = (a1 , . . . , an ). The point La can be written as the sum j � La = aj Lej , and L can be written out in components as L = (L1 , . . . , Lm ), where each Lj : Rn → R is a linear map. Then Lej = (L1 , ej , . . . , Lm ej ), and Li ej = �i,j . The numbers �i,j form an n × n matrix denoted by [�i,j ]. Remark. Let U ⊆ Rn , and let f1 : Rn → Rm1 and f2 : Rn → Rm2 be differentiable maps. Let m = m1 + m2 , so that Rm1 × Rm2 = Rm . Now, construct a function f : Rn → Rm defined in component form by f = (f1 , f2 ). The derivative of f at a is Df (a) = (Df1 (a), Df2 (a)). (2.21) Remark. Let f : U → Rm be a map. The action of f on input x written out in component form is f (x) = (f1 (x), . . . , fm (x)). So, the map can be represented in component form as f = (f1 , . . . , fm ), where each fi as a map of the form fi : U → R. The derivative of f acting on the standard basis vector ej is Df (a)ej = (Df1 (a)ej , . . . , Dfm (a)ej ) ∂fm ∂f1 (a), . . . , (a)). =( ∂xj ∂xj So, the derivative (Df )(a) can be represented by an m × n matrix � � ∂fi (Df )(a) ∼ Jf (a) = (a) = ∂xj called the Jacobian matrix of f at a, which you probably recognize. 4 (2.22) (2.23)...
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This note was uploaded on 10/02/2010 for the course MAT unknown taught by Professor Unknown during the Fall '04 term at MIT.

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