Engineering Calculus Notes 443

Engineering Calculus Notes 443 - 431 4.2. DIFFERENTIABLE...

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Unformatted text preview: 431 4.2. DIFFERENTIABLE MAPPINGS and the Jacobian of their composition is [(f ◦ g)′ (x0 )] = J (f ◦ g)(x0 ) = Jf (y0 ) · Jg(x0 ) = [f ′ (y0 ) g′ (x0 )]. → → Second, if − : R → R3 is a parametrization − (t) of the curve C and p p → → 3 → R is a function defined on C , then (f ◦ − )(t) = f (− (t)) gives f as a f: R p p →→ − = − (t), function of the parameter t: letting x p → → Jf (− ) = [d− f ] x x ∂ f ∂f ∂x ∂y ′ x → − (t) = y ′ Jp z′ = and ∂f ∂z d → → [f (− (t))] = J (f ◦ − )(t) p p dt → → = Jf (− ) · J − (t) x p = ∂ f ′ ∂f ′ ∂f ′ x+ y+ z. ∂x ∂y ∂z → → → Third, if again − : R → R3 and − : R → R3 are two parametrizations (− (t) p q p →(s)) of the curve C , and τ : R → R is the change-of-parameter function − and q → → →→ t = τ (s) (i.e., − (s) = − (τ (s)), or − = − ◦ τ ), then q p q p dx dt dx → J − ( s) = q ds dz dt dz ds dt J τ (s) = τ ′ (s) = ds → J − (t) = p dy dt dy ds and → → J − (s) = J (− ◦ τ )(s) q p − (t) · Jτ (s) → =Jp = dx dt dt ds dy dt dt ds dz dt dt ds ...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Spring '08 term at University of Florida.

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