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Laplacian, partial derivatives

  1. Aug 3, 2010 #1
    1. The problem statement, all variables and given/known data
    Find the Laplacian of F = [tex]sin(k_x x)sin(k_y y)sin(k_z z)[/tex]


    2. Relevant equations

    [tex]\nabla^2 f = \left( \frac{\partial}{\partial x} +\frac{\partial}{\partial y} + \frac{\partial}{\partial z} \right) \cdot \left( \frac{\partial}{\partial x} + \frac{\partial}{\partial y} + \frac{\partial}{\partial z} \right) \cdot F[/tex]
    Where F is a scalar function

    3. The attempt at a solution

    Biggest problem is with partial derivatives. I don't know how to approach taking a partial derivative of such a big multivariate product :( Just got the derivative of sin(x) is cos(x) and the second derivative -sin(x)

    Just want to makes sure, but [tex]\frac{\partial}{\partial x}y = 0 [/tex] but [tex]\frac{\partial}{\partial x}xy = y [/tex] ?
     
  2. jcsd
  3. Aug 3, 2010 #2

    Dick

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    Homework Helper

    Yes to everything you've written. The dot of the two vector operators is the sum of the second derivatives of F. There is no dot product with F. F is a scalar. What do you get if you add the three second derivatives of F with respect to x, y and z?
     
  4. Aug 3, 2010 #3
    It will just be the sum that is the Laplacian.. as you just said :O

    Problem is how to do one partial derivative, for starters, on
    [tex]
    sin(k_x x)sin(k_y y)sin(k_z z)
    [/tex]
     
  5. Aug 4, 2010 #4

    HallsofIvy

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    What is the derivative, with respect to x, of [itex]AB sin(k_x x)[/itex] with A and B constants?

    What is the derivative, with respect to y, of [itex]AB sin(k_y y)[/itex] with A and B constants?

    What is the derivative, with respect to z, of [itex]AB sin(k_z z)[/itex] with A and B constants?
     
  6. Aug 9, 2010 #5
    I think when you're taking the partial derivative of the first term, the k term gets pulled out by the chain rule, then just change it to cos. Taking the second derivative brings the k out again by the chain rule, leaving k^. So it will be -k^2(sin-sin-sin)
     
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