Question

Let
r = x i + y j + z k
Find each of the following. (Give your answers in terms of r.)

∇ · (|r| r)

Answer 1

∇ · (|r| r) =|r| ∇ · ( r) + r ∇ · |r|

|r| =√x²+y²+z²
∇ · ( r) =dr/dx +dr/dy+dr/dz= i+j+k
so
∇ · (|r| r) =√x²+y²+z² (i+j+k) + (x i + y j + z k ) ∇ · |r|


∇ · |r|= 2(x+y+z)/√x²+y²+z²
finally

∇ · (|r| r) = |r| (i+j+k) + 2(xi+yj+zk)/ |r|=|r| (i+j+k) + 2r/|r|