Engineering Calculus Notes 445

Engineering Calculus Notes 445 - 4.2. DIFFERENTIABLE...

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Unformatted text preview: 4.2. DIFFERENTIABLE MAPPINGS with Jacobian J (P ol)(r, θ ) = cos θ −r sin θ sin θ r cos θ and the Chain Rule tells us that ∂ m ∂m = J (f ◦ P ol)(r, θ ) ∂r ∂θ = Jf (x, y ) · J (P ol)(r, θ ) ∂ f ∂f cos θ −r sin θ · sin θ r cos θ ∂x ∂y ∂f ∂f ∂f ∂f cos θ + sin θ − r sin θ + r cos θ = ∂x ∂y ∂x ∂y = in other words, ∂m ∂m ∂x ∂m ∂y = + ∂r ∂x ∂r ∂y ∂r ∂f ∂f + (sin θ ) = (cos θ ) ∂x ∂y and ∂m ∂m ∂x ∂m ∂y = + ∂θ ∂x ∂θ ∂y ∂θ ∂f ∂f + (r cos θ ) . = (−r sin θ ) ∂x ∂y For example, if y m = f (x, y ) = x then m= r sin θ = tan θ ; r cos θ using the Chain Rule, we have ∂m ∂f ∂x ∂f ∂y = + ∂r ∂x ∂r ∂y ∂r 1 y = − 2 (cos θ ) + (sin θ ) x x sin θ −r sin θ cos θ + = 2 cos2 θ r r cos θ =0 433 (4.3) ...
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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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