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Homework Help: Partial differentiation

  1. Dec 14, 2013 #1

    patrickmoloney

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    1. The problem statement, all variables and given/known data

    let V=f(x²+y²) , show that x(∂V/∂y) - y(∂V/∂x) = 0


    2. Relevant equations




    3. The attempt at a solution

    V=f(x²+y²) ; V=f(x)² + f(y)²

    ∂V/∂x = 2[f(x)]f'(x) + [0]

    ∂V/∂y = 2[f(y)]f'(y)

    I'm sure I've gone wrong somewhere, I have never seen functions like this, I'm just used to using V=f(x,y)= some function and then partially differentiations. help would be much apprectiated.
     
  2. jcsd
  3. Dec 14, 2013 #2

    Simon Bridge

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    You are nearly there:
    You have a function of form: ##V(x,y)=f(g(x,y))##
    ... so you'd use the chain rule. $$\partial_x V = \frac{df}{dg}\partial_x g(x,y)$$
     
  4. Dec 14, 2013 #3

    patrickmoloney

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    V(x,y) = f(g(x,y)

    using chain rule:

    ∂V/∂x = df/dg (∂g/∂x[(x²+y²)])
    = 2x(df/dg)

    y(∂V/∂x) = 2xy(df/dg)

    ∂V/∂y = df/dg (∂g/∂y[(x²+y²)])
    = 2y(df/dg)

    x(∂V/∂y) = 2xy(df/dg)

    x(∂V/∂y)-y(∂V/∂x) = 0

    2xy(df/dg) - 2xy(df/dg) = 0
     
  5. Dec 14, 2013 #4

    Simon Bridge

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    Well done :)
     
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