Lec09 - Math 1920 Lecture addendum Given a multivariable...

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Math 1920 - Feb 11, 2009 Lecture addendum Given a multivariable function f ( x ( r, θ ) , y ( r, θ )). In the example of polar coordinates, x ( r, θ ) = r cos θ and y ( r, θ ) = r sin θ . 1. Use the chain rule to find ∂f ∂r . ∂f ∂r = ∂f ∂x ∂x ∂r + ∂f ∂y ∂y ∂r = f x ∂x ∂r + f y ∂y ∂r 2. Use the chain rule to find ∂f ∂θ . ∂f ∂θ = ∂f ∂x ∂x ∂θ + ∂f ∂y ∂y ∂θ = f x ∂x ∂θ + f y ∂y ∂θ 3. Use the chain rule to find 2 f ∂r 2 . (hint: use part 1) 2 f ∂r 2 = ∂r ∂f ∂r = ∂r ∂f ∂x ∂x ∂r + ∂f ∂y ∂y ∂r = ∂r ∂f ∂x ∂x ∂r + ∂f ∂x ∂r ∂x ∂r + ∂r ∂f ∂y ∂y ∂r + ∂f ∂y ∂r ∂y ∂r = ∂x ∂f ∂x ∂x ∂r + ∂y ∂f ∂x ∂y ∂r ∂x ∂r + 2 x ∂r 2 ∂f ∂x + ∂x ∂f ∂y ∂x ∂r + ∂y ∂f ∂y ∂y ∂r ∂y ∂r + 2 y ∂r 2 ∂f ∂y = 2 f ∂x 2 ∂x ∂r 2 + 2 f ∂y∂x ∂y ∂r ∂x ∂r + 2 x ∂r 2 ∂f ∂x + 2 f ∂x∂y ∂x ∂r ∂y ∂r + 2 f ∂y 2 ∂y ∂r 2 + 2 y ∂r 2 ∂f ∂y = 2 f ∂x 2 ∂x ∂r 2 + 2 f ∂y 2 ∂y ∂r 2 + 2 2 f ∂y∂x ∂y ∂r ∂x ∂r + 2 x ∂r 2 ∂f ∂x + 2 y ∂r 2 ∂f ∂y 4. Use the chain rule to find 2 f ∂θ 2 . (hint: use part 2) 2 f ∂θ 2 =
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