Question

Let r = x i + y j + z k and r = |r|. If F = r/r p, find div F. (Enter your answer in terms of r and p.) div F =

Answer 1

Answer:

The answer in terms of r and p is div F = (3 - p)/r^p.

Step-by-step explanation:

Given:

F = R/r^p

F= <x, y, z> / |<x, y, z>|^p

F = <x/(x^2+y^2+z^2)^(p/2), y/(x^2+y^2+z^2)^(p/2), z/(x^2+y^2+z^2)^(p/2)>.

Hence,  For div F ,

We take partial derivative:

div F  = (∂/∂x) x/(x^2+y^2+z^2)^(p/2) + (∂/∂y) y/(x^2+y^2+z^2)^(p/2) + (∂/∂z) z/(x^2+y^2+z^2)^(p/2)

Now, we use the rational derivative rule to find the derivatives:

div F  = [1(x^2+y^2+z^2)^(p/2) - x * px(x^2+y^2+z^2)^(p/2 - 1)] / (x^2+y^2+z^2)^p  + [1(x^2+y^2+z^2)^(p/2) - y * py(x^2+y^2+z^2)^(p/2 - 1)] / (x^2+y^2+z^2)^p  + [1(x^2+y^2+z^2)^(p/2) - z * pz(x^2+y^2+z^2)^(p/2 - 1)] / (x^2+y^2+z^2)^p

div F  = (x^2+y^2+z^2)^(p/2 - 1) {[(x^2+y^2+z^2) - px^2] + [(x^2+y^2+z^2) - py^2] + [(x^2+y^2+z^2) - pz^2]}  / (x^2+y^2+z^2)^p

div F  = [3(x^2+y^2+z^2 - p(x^2+y^2+z^2)] / (x^2+y^2+z^2)^(p/2 + 1)

div F  = (3 - p) (x^2+y^2+z^2) / (x^2+y^2+z^2)^(p/2 + 1)

div F  = (3 - p)/(x^2+y^2+z^2)^(p/2)

Now it comes like,

div F  = (3 - p)/r^p.