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Chain rule

  1. Sep 8, 2013 #1
    I am given Z = f (x, y), where x= r cosθ and y=r sinθ

    I found

    ∂z/∂r = ∂z/∂x ∂x/∂r + ∂z/∂y ∂y/∂r = (cos θ) ∂z/∂x + (sin θ) ∂z/∂y and

    ∂z/∂θ = ∂z/∂x ∂x/∂θ + ∂z/∂y ∂y/∂θ= (-r sin θ) ∂z/∂x + (r cos θ) ∂z/∂y

    I need to show that

    ∂z/∂x = cos θ ∂z/∂r - 1/r * sin θ ∂z/∂θ and

    ∂z/∂y = sin θ ∂z/∂r + 1/r * cos θ ∂z/∂θ

    Any ideas?

  2. jcsd
  3. Sep 8, 2013 #2


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    ∂z/∂r = (cos θ) ∂z/∂x + (sin θ) ∂z/∂y

    ∂z/∂θ = (-r sin θ) ∂z/∂x + (r cos θ) ∂z/∂y

    It is two equations for the "unknowns" ∂z/∂x and ∂z/∂y. Multiply the first equation by r(cosθ), the second one with (sinθ) and subtract them. What do you get?

    Last edited: Sep 8, 2013
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