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Partial derivatives extensive use

  1. Nov 2, 2012 #1
    1. The problem statement, all variables and given/known data


    let u be a function of x and y.using x=rcosθ y=rsinθ,transform the following expressions in the terms of partial derivatives with respect to polar coordinates:(d^u/dx^2(double derivative of u with respect to x)+d^2u/dy^2(double derivative of u with respect to y)
    2. Relevant equations
    chain rule in partial derivatives


    3. The attempt at a solution
    first i differentiated u with respect to theta by using chain rule and then with respect to r also by using chain rule.first has no r term wheras 2nd has r terms so no way the terms can cancel also.please tell me how to proceed or method i should use to solve this problem
     
  2. jcsd
  3. Nov 3, 2012 #2

    Mark44

    Staff: Mentor

    Are these the partials you need to find?

    $$ \frac{\partial^2 u}{\partial r^2}~ \text{and}~\frac{\partial^2 u}{\partial^2 \theta}$$

    If so, what did you get for these first partials?
    $$ \frac{\partial u}{\partial r}$$
    $$ \frac{\partial u}{\partial \theta}$$

    For problems like this I find it helpful to draw a diagram of the relationships between all the variables.
    Code (Text):

          x ------ θ
     /
    u
      \  y -------r
     
    Although I can't show them, there are also lines between x and r and between y and θ.

    The idea is that there are two ways to get from u to θ (through x and y), and there are two ways to get from u to r (also through x and y). This helps to get across the idea that each partial involves the sum of two terms.

    Using subscripts to indicate partials, and letting u = f(x, y), we have
    uθ = fx * xθ + fy * yθ, and
    ur = fx * xr + fy * yr

    To get the second partials (we don't call them double partials), you need to differentiate uθ with respect to θ, and differentiate ur with respect to r.

     
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