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# 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/(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)}>.

So, For div F,

Then, we take partial derivative:

div F = (d/dx) x/(x^{2} + y^{2} + z^{2})^{(p/2)} + (d/dy) y/(x^{2} + y^{2} + z^{2})^{(p/2)} + (d/dz) z/(x^{2} + y^{2} + z^{2})^{(p/2)}

Now, we have to 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}.

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}.

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