# Gradient of |xi + yj + zk|^-n

1. Mar 21, 2009

### sci-doo

1. The problem statement, all variables and given/known data
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!

2. Mar 21, 2009

### 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.