Calculating the Gradient of a Vector Function with a Power Function

[sci-doo]

Homework Statement

Let f(x,y,z)= |r|^-n where r = x\hat{i} + y\hat{j} + z\hat{k}

Show that

\nabla 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

\nabla 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!

 

[yyat]

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 |r|=\sqrt{x^2+y^2+z^2}, then compute the partials.