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.)

Solution:

It is given that, r = x i + y j + z k and r = |r|.

F = R/r^p

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

F = <x/(x² + y² + z²)^(p/2), y/(x² + y² + z²)^(p/2), z/(x² + y² + z²)^(p/2)>.

So, For div F,

Then, we take partial derivative:

div F = (d/dx) x/(x² + y² + z²)^(p/2) + (d/dy) y/(x² + y² + z²)^(p/2) + (d/dz) z/(x² + y² + z²)^(p/2)

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

div F = [1(x² + y² + z²)^(p/2) - x × px(x² + y² + z²)^{(p/2 - 1)}] / (x² + y² + z²)^p + [1(x² + y² + z²)^(p/2) - y × py(x² + y² + z²)^{(p/2 - 1)}] / (x² + y² + z²)^p + [1(x² + y² + z²)^(p/2) - z × pz(x² + y² + z²)^{(p/2 - 1)}] / (x² + y² + z²)^p

div F = (x² + y² + z²)^{(p/2 - 1)}{[(x² + y² + z²) - px²] + [(x² + y² + z²) - py²] + [(x² + y² + z²) - pz²]} /(x² + y² + z²)^p

div F = [3(x² + y² + z² - p(x² + y² + z²)] / (x² + y² + z²)^{(p/2 + 1)}

div F = (3 - p) (x² + y² + z²) / (x² + y² + z²)^{(p/2 + 1)}

div F = (3 - p)/(x² + y² + z²)^(p/2)

Now it comes like,

div F = (3 - p)/r^p.

Therefore, div F = (3 - p)/r^p.

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.)

Summary:

Let r = x i + y j + z k and r = |r|. If F = r/r^p, div F = (3 - p)/r^p.