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Math Help - Change of Coordinates

  1. #1
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    Change of Coordinates

    If f(x,y) becomes g(r,t) in a transformation from Cartesian to Polar cooardinates, show that

    i) \left(\frac{\partial f}{\partial x}\right)^2 + \left(\frac{\partial f}{\partial y}\right)^2= \left(\frac{\partial g}{\partial r}\right)^2 + \frac{1}{r^2}\left(\frac{\partial g}{\partial t}\right)^2

    ii) \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = \frac{\partial^2 g}{\partial r^2}+\frac{1}{r}\frac{\partial^2 g}{\partial r} + \frac{1}{r^2}\frac{\partial^2 g}{\partial t^2}

    I know the partials of x and y with respect to r and t and vice-versa; I just can't get my head around how to approach the question.

    I know how to find \frac{\partial f}{\partial r} and \frac{\partial f}{\partial t} with the chain rule, and same for g, but not the other way round. Help!
    Last edited by harbottle; October 15th 2009 at 03:34 AM.
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  2. #2
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    Quote Originally Posted by harbottle View Post
    If f(x,y) becomes g(r,t) in a transformation from Cartesian to Polar cooardinates, show that

    i) \left(\frac{\partial f}{\partial x}\right)^2 + \left(\frac{\partial f}{\partial y}\right)^2= \left(\frac{\partial g}{\partial r}\right)^2 + \frac{1}{r^2}\left(\frac{\partial g}{\partial t}\right)^2

    ii) \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = \frac{\partial^2 g}{\partial r^2}+\frac{1}{r}\frac{\partial^2 g}{\partial r} + \frac{1}{r^2}\frac{\partial^2 g}{\partial t^2}

    I know the partials of x and y with respect to r and t and vice-versa; I just can't get my head around how to approach the question.

    I know how to find \frac{\partial f}{\partial r} and \frac{\partial f}{\partial t} with the chain rule, and same for g, but not the other way round. Help!

    So x = r cos t, y = r sin t and then g(r,w) = f(x,y){under change) = f(r cos t, r sin t), so:

    dg/dr = (df/dx)(dx/dr) + (df/dy)(dy/dr)= (df/dx)cos t + (df/dy)sin t

    dg/dt = (df/dx)(-r sin t) + (df/dy)(rcos t)

    Now just square above, add both lines and make some order in that mess and you'll get what you want, both (i) and (ii)

    Tonio
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  3. #3
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    So we just use the fact that g=f.. simple.

    thank you!
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