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.
