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Homework Help: Gradient of |xi + yj + zk|^-n

  1. Mar 21, 2009 #1
    1. The problem statement, all variables and given/known data
    Let f(x,y,z)= |r|-n where r = x[tex]\hat{i}[/tex] + y[tex]\hat{j}[/tex] + z[tex]\hat{k}[/tex]

    Show that

    [tex]\nabla[/tex] f = -nr / |r|n+2

    2. The attempt at a solution
    Ok, I don't care about the absolute value (yet at least).

    I take partial derivatives of (xi + yj + zk)^-n and get

    [tex]\nabla[/tex] f = i(-n)(xi + yj + zk)^(-n-1) + j(-n)(xi + yj + zk)^(-n-1) + k(-n)(xi + yj + zk)^(-n-1)

    = -n(i + j + k)*(xi + yj + zk)^(-n-1)

    But according to problem statement what I should get is:
    -nr / |r|n+2 = -n (i x + j y + k z)^(-1 - n)

    I don't understand where the (i + j + k) term goes! :eek:
  2. jcsd
  3. Mar 21, 2009 #2
    The | | does not refer to the absolute value, but the norm in this case. In fact xi+yj+zk is a vector and it does not make sense to compute the power of a vector.

    So what you have to use is that [tex]|r|=\sqrt{x^2+y^2+z^2}[/tex], then compute the partials.
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