Engineering Calculus Notes 241

Engineering Calculus Notes 241 - 229 3.2. LINEAR AND AFFINE...

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Unformatted text preview: 229 3.2. LINEAR AND AFFINE FUNCTIONS → Remark 3.2.2. Given any − 0 ∈ R3 , the affine function φ: R3 → R can be x → x written in the form of Equation (3.1), as its value at − 0 plus a linear → function of the displacement from − 0 : x → → → → φ(− 0 + △− ) = φ(− 0 ) + ℓ(△− ) x x x x or, stated differently, → → → → φ(− 0 + △− ) − φ(− 0 ) = ℓ(△− ) ; x x x x → → the displacement of φ(− ) from φ(− 0 ) is a linear function of the x x → − from − . → displacement of x x0 In light of this observation, we can use Remark 3.2.1 to determine an affine → function from its value at a point − 0 together with its values at the points x → − + − obtained by displacing the original point in a direction parallel to → x0 ej one of the coordinate axes. A brief calculation shows that → → φ(− 0 + △− ) = a0 + x x 3 j =1 aj △ x j (3.2) where → △ − = ( △ x1 , △ x2 , △ x3 ) x → a = φ(− ) x 0 0 and for j = 1, 2, 3 → → → aj = φ(− 0 + − j ) − φ(− 0 ) . x e x Exercises for § 3.2 Practice problems: → 1. For each linear function ℓ(− ) below, you are given the values on the x standard basis. For each function, find ℓ(1, −2, 3) and ℓ(2, 3, −1). → − → → (a) ℓ(− ) = 2, ℓ(− ) = −1, ℓ k = 1. ı → − → → (b) ℓ(− ) = 1, ℓ(− ) = 1, ℓ k = 1. ı ...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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