Engineering Calculus Notes 444

Engineering - 432 CHAPTER 4 MAPPINGS AND TRANSFORMATIONS in other words q ′(s = p ′(t dt ds The second and third examples above have further

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Unformatted text preview: 432 CHAPTER 4. MAPPINGS AND TRANSFORMATIONS in other words, q ′ (s) = p ′ (t) dt . ds The second and third examples above have further generalizations in light of Theorem 4.2.5. → If f : R3 → R is a function defined on the surface S and − (s, t) is a p → − : R2 → R3 ), then f ◦ − expresses f as a function of → parametrization of S ( p p the two parameters s and t, and the Chain Rule gives the partials of f →→ with respect to them: setting − = − (s, t), x p so ∂ f ∂f ∂f → Jf (− ) = x ∂x ∂y ∂z ∂x/∂s ∂x/∂t → J − (s, t) = ∂y/∂s ∂y/∂t p ∂z/∂s ∂z/∂t → → → J (f ◦ − )(s, t) = Jf (− ) · J − (s, t) p x p = ∂ f ∂x ∂f ∂y ∂f ∂z + + ∂x ∂s ∂y ∂s ∂z ∂s ∂f ∂x ∂f ∂y ∂f ∂z + + ; ∂x ∂t ∂y ∂t ∂z ∂t the first entry says ∂f ∂f ∂x ∂f ∂y ∂f ∂z = + + ∂s ∂x ∂s ∂y ∂s ∂z ∂s while the second says ∂f ∂x ∂f ∂y ∂f ∂z ∂f = + + . ∂t ∂x ∂t ∂y ∂t ∂z ∂t We can think of changing coordinates for a function of two variables as the analogue of this when S is the xy -plane. In this case we drop the third variable. In particular, if a measurement is expressed as a function m = f (x, y ) of the rectangular coordinates, then its expression in terms of polar coordinates is (f ◦ (P ol))(r, θ ), where P ol: R2 → R2 is the change-of-coordinates map P ol(r, θ ) = r cos θ r sin θ ...
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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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