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Math Help - chain rule in complex numbers

  1. #1
    Junior Member rubix's Avatar
    Joined
    Apr 2009
    Posts
    49

    chain rule in complex numbers

    f(z) = u + iv<br /> <br />
u = u(x,y)<br />
v = v(x,y)

    use chain rule and
    x = rcos (\theta)<br />
y = r sin (\theta)

    to show

    u_x = u_r * cos (\theta) - u_\theta * sin(\theta)/r

    v_x  = v_r * cos(\theta) - v_\theta * sin(\theta)/r

    =============
    what i know is, if i had to find u_r then i would go about:

    u_r = u_x * u_r + u_y * y_r

    plugging values gives

     u_r = u_x * cos(\theta) + u_y * sin(\theta)
    similarly can find [/tex] u_\theta , v_r and v_theta[/tex]

    but to find u_x, how would i set up the chain rule to begin with? In above case, x and y are function of r and theta so it makes sense. ho.wever, here i can't do similar with u_x...i think.


    please show full work if possible.
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  2. #2
    Senior Member
    Joined
    Jan 2010
    Posts
    354
    For this problem, consider that you could also define r,\theta as functions of x,y, like so:

    r=\sqrt{x^2+y^2}

    \theta = \tan^{-1}\left( \frac{y}{x} \right)

    Then the chain rule looks like this:

    u_x = u_r  r_x + u_\theta  \theta_x

    Think you can finish it from here?
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