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Chain rule for partial derivatives

  1. Mar 18, 2010 #1
    If I have u = u(x,y) and let (r, t) be polar coordinates, then
    expressing u_x and u_y in terms of u_r and u_t could be
    done with a system of linear equations in u_x and u_y?

    I get:
    u_r = u_x * x_r + u_y * y_r

    u_t = u_x * x_t + u_y * y_t

    is this the right direction? Because by substitution,
    I end up with:
    (u_t* x_r - u_r)/(y_t*x_r - y_r) = u_y which
    does not seem right, considering:
    u_y = u_r * r_y + u_t * t_y

    Any insight to the problem is appreciated.
  2. jcsd
  3. Mar 19, 2010 #2
    No, that should be

    (u_t x_r - u_r x_t ) / (y_t x_r - x_t y_r) = u_y

    and since

    x_r = cos t
    y_r = sin t

    x_t = -r sin t
    y_t = r cos t

    it implies that u_y = (u_t cos t + u_r r sin t) / (r cos t cos t + r sin t sin t) = u_t cos t / r + u_r sin t
  4. Mar 20, 2010 #3
    I found my error. Thanks for your response!

    One question:
    How did you determine the following:
    x_t = -r sin t
    y_t = r cos t
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