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Homework Help: Difficulty with the Laplacian

  1. Jun 23, 2015 #1
    1. The problem statement, all variables and given/known data
    Given: |r|=√(x^2+y^2+z^2) r=xi+yj+zk

    (i)Find the partial derivative with respect to x of |r|.
    (ii) Find the Laplacian of |r|.

    2. Relevant equations

    3. The attempt at a solution
    For (i) I got x/|r|
    but then for (ii) I got 2/r which I don't think is correct
  2. jcsd
  3. Jun 23, 2015 #2


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    If ##| \vec r(x, y, z) | = \sqrt{x^2 + y^2 + z^2}##, then:

    $$| \vec r(x, y, z) |_x = \frac{\partial}{\partial x} (x^2 + y^2 + z^2)^{\frac{1}{2}} = \frac{1}{2} (x^2 + y^2 + z^2)^{- \frac{1}{2}} \cdot \frac{\partial}{\partial x} (x^2 + y^2 + z^2)$$

    What is the definition of the Laplacian?
  4. Jun 23, 2015 #3


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    Part i) is the warm up for part ii).
    What did you do to get x/|r|?

    If ##\frac{\partial}{\partial x } |r| = \frac{x}{|r|} ##, then what is ##\frac{\partial}{\partial x } \frac{x}{|r|} ##?

    I think 2/|r| is right.
  5. Jun 23, 2015 #4


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    If you clean up the computation in the second post:

    $$\frac{1}{2} (x^2 + y^2 + z^2)^{- \frac{1}{2}} \cdot \frac{\partial}{\partial x} (x^2 + y^2 + z^2) = \frac{x}{\sqrt{x^2 + y^2 + z^2}} = \frac{x}{| \vec r |}$$

    It is, but it would be nice if the OP showed some of the work.
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