The discussion focuses on calculating the partial derivative ∂f/∂x for the function f(r,θ) = rsin²(θ) using the chain rule. The variables are defined as x = rcosθ and y = rsinθ. The correct expression for ∂f/∂x is derived as ∂f/∂x = -xy² / (x² + y²)^(3/2), aligning with the answer provided in the textbook. Participants emphasize the importance of expressing r and θ as functions of x and y to accurately compute the derivatives.
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To find ∂r/∂x and ∂θ/∂x, it's helpful to have r as a function of x and y, and θ as a function of x and y.Sheldinoh said:Homework Statement
Find: ∂f/∂x
f(r,θ)=rsin^2(θ), x=rcosθ, y=rsinθ
The Attempt at a Solution
∂f/∂r=sin^2θ ∂r/∂x=-cosθ/x
∂f/∂θ=2*r*cosθ*sinθ ∂θ/∂x=-1/sqrt(1-(x^2)/(r^2)
To find ∂r/∂x and ∂θ/∂x, you need r as a function of x and y, and θ as a function of x and y.Sheldinoh said:∂f/∂x = -sin^2θcosθ/x^2 + -2*r*cosθ*sinθ/sqrt(1-(x^2)/(r^2)
∂f/∂x = -y-sqrt(x^2+y^2) / (x^2+y^2)^3/2
THE ANSWER IN THE BACK OF THE BOOK IS :
∂f/∂x = -xy^2 / (x^2+y^2)^3/2