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Thread: chain rule in complex numbers

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

    chain rule in complex numbers

    $\displaystyle f(z) = u + iv

    u = u(x,y)
    v = v(x,y)$

    use chain rule and
    $\displaystyle x = rcos (\theta)
    y = r sin (\theta)$

    to show

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

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

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

    $\displaystyle u_r = u_x * u_r + u_y * y_r$

    plugging values gives

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

    but to find $\displaystyle u_x$, 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 $\displaystyle r,\theta$ as functions of $\displaystyle x,y$, like so:

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

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

    Then the chain rule looks like this:

    $\displaystyle u_x = u_r r_x + u_\theta \theta_x$

    Think you can finish it from here?
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