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Differentiating a polar function

  1. Mar 17, 2012 #1
    1. The problem statement, all variables and given/known data
    let z=f(x,y) be a differentiable function. If we change to polar coordinates, we make the substitution x=rcos(θ), y=rsin(θ), x^2+y^2=r^2 and tan(θ) = y/x.
    a. Find expressions ∂z/∂r and ∂z/∂θ involving ∂z/∂x and ∂z/∂y.
    b. Show that (∂z/∂x)^2 + (∂z/∂y)^2 = (∂z/∂r)^2 + (1/r^2)(∂z/∂θ)^2.


    3. The attempt at a solution

    a. i understand that f(x,y) in polar is f(r,θ) but don't understand how to calculate the partial derivatives of ∂z/∂x and ∂z/∂y because there is not know function for z...
     
  2. jcsd
  3. Mar 17, 2012 #2

    HallsofIvy

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    Do you know the chain rule for functions of two variables?
     
  4. Mar 17, 2012 #3
    in the specific case of this problem they come out like this:
    ∂z/∂r = (∂z/∂x)(∂x/∂r) + (∂z/∂y)(∂y/∂r)
    ∂z/∂θ = (∂z/∂x)(∂x/∂θ ) + (∂z/∂y)(∂y/∂θ)

    right?
     
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