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Partial differentiation: prove this general result

  1. Dec 31, 2012 #1
    1. The problem statement, all variables and given/known data

    The function f(x,y,z) may be expressed in new coordinates as g(u,v,w). Prove this general result:

    2v337fd.png



    3. The attempt at a solution

    df = (∂f/∂x)dx + (∂f/∂y)dy + (∂f/∂z)dz

    dg = (∂g/∂u)du + (∂g/∂v)dv + (∂g/∂w)dw


    df = dg since they are the same thing?

    but the components dx, dy, dz, du, dv, dw are different so i can't equate them...
     
  2. jcsd
  3. Dec 31, 2012 #2

    pasmith

    User Avatar
    Homework Helper

    Yes. Here x, y and z are to be regarded as functions of u, v and w, so you need to work out what dx, dy and dz are in terms of du, dv and dw and substitute those into the equation for df.
     
  4. Jan 1, 2013 #3
    how can we assume that x = x(u,v,w) and y = y(u,v,w) and z = z(u,v,w)?
     
  5. Jan 1, 2013 #4
    Since f(x,y,z) may be expressed in new coordinates as g(u,v,w), it means that the function takes a different form when x, y and z are replaced by u, v and w. For example, the function z(x,y)=x+iy (rectangular) becomes t(r, θ)=r(cosθ+isinθ) (polar) when x is replaced by rcosθ and y is replaced by rsinθ. Thus x and y are both functions of r and θ.
     
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