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 =

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 = / ||^p

F =.

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.