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Multivariable Calc: ∇ ∙ (r^3 * ȓ )

  1. Oct 18, 2015 #1
    1. The problem statement, all variables and given/known data

    Evaluate: ∇ • (r^3 * ȓ ) (del dot (r^3 times vector r)), where r = sqrt(x^2+y^2+z^2) and vector r = (x, y, z)

    3. The attempt at a solution

    So, taking the partial derivative of the x component, I got 2r3/2. Doing the same thing for the y, z components I got a similar answer. This gives me a solution of ∇ • (r^3 * ȓ ) = 6r3/2. However, according to my professor, the actual answer is just 6r^3. I don't know where I'm going wrong in my calculations, and any and all help would be appreciated. Thank you!
     
  2. jcsd
  3. Oct 18, 2015 #2
    Try the del operator in spherical coordinates.
     
  4. Oct 18, 2015 #3
    I thought about doing that, but my professor always uses r = ||r|| where r is a vector (x, y, z), and I've used this same definition of r previously in the same worksheet to get the right answer. However, if I were to do that, how would I set that problem up?
     
  5. Oct 18, 2015 #4

    Ray Vickson

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    You have the wrong derivatives:
    [tex] \frac{\partial}{\partial x} x r^3 = \frac{\partial}{\partial x} x (x^2+y^2+z^2)^{3/2} \neq 2 r^{3/2} = 2 (x^2+y^2+z^2)^{3/4} [/tex]
     
  6. Oct 18, 2015 #5
    I'm sorry, but how did you get that? I get to [itex] (x^2+y^2+z^2)^{1/2}(4x^2+y^2+z^2) [/itex] and I don't understand how you simplified that to [itex] 2(x^2+y^2+z^2)^{3/4}[/itex]
     
  7. Oct 18, 2015 #6

    Ray Vickson

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    No: that is what YOU wrote, and I am saying it is wrong. You wrote ##2 r^{3/2}## and that equals ##2 (x^2+y^2+z^2)^{3/4}##, because ##r = (x^2+y^2+z^2)^{1/2}##.
     
  8. Oct 18, 2015 #7
    Wow. I completely misread that haha. Sorry about that. So, I probably simplified [itex] (x^2+y^2+z^2)^{1/2}(4x^2+y^2+z^2) [/itex] incorrectly, but I think that's the correct derivative. If so, is there a way to manipulate the expression so I get 2r?
     
  9. Oct 19, 2015 #8

    Ray Vickson

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    No, but you don't need to do that.
     
  10. Oct 19, 2015 #9

    ehild

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    You got the derivative with respect to x. But you need the divergence of the given function. How is divergence defined? What product of the vector-vector function with the nabla operator?
     
  11. Oct 19, 2015 #10
    @Ray Vickson, @ehild: I actually did the whole problem instead of just one part and got the right answer. Who'dathunk? I guess I got caught up in just the first part. This is what happens when the right answer is directly underneath the problem haha. Thanks for helping me out! :smile:
     
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