math

If z=r^2 tan^2(theta),x=r cos(theta),y=r sin(theta),
find the {partial^2 z}/{partial(x)partial(y)}

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asked by Jiang
  1. staying with parametric functions makes it hard.
    So, let
    r^2 = x^2+y^2
    tanθ = y/x
    z = r^2 tan^2θ = (x^2+y^2)(y^2/x^2) = y^2 + y^4/x^2
    ∂z/∂x = -2y^4/x^3
    ∂z/∂y = 2y + 4y^3/x^2
    ∂^2z/∂x∂y = ∂^2z/∂y∂x = -8y^3/x^3 = -8tan^3θ

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    posted by oobleck

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