- #1

ineedmunchies

- 45

- 0

## Homework Statement

**F**= [tex]\frac{

**r**}{r}[/tex]

Find div

**F**

and curl

**F**

## Homework Equations

**r**= x[tex]\widehat{i}[/tex] + y[tex]\widehat{j}[/tex] + z[tex]\widehat{k}[/tex]

r = [tex]\sqrt{(x^{2} + y^{2} + z^{2})}[/tex]

## The Attempt at a Solution

**F**= [tex]\frac{x}{(\sqrt{x^{2} + y^{2} + z^{2}})}[/tex][tex]\widehat{i}[/tex] + [tex]\frac{y}{(\sqrt{x^{2} + y^{2} + z^{2}})}[/tex][tex]\widehat{j}[/tex] + [tex]\frac{z}{(\sqrt{x^{2} + y^{2} + z^{2}})}[/tex][tex]\widehat{k}[/tex]

div

**F**= [tex]\frac{\partial}{\partial x}[/tex] [tex](x(x^{2} + y^{2} + z^{2})^{\frac{-1}{2}}[/tex] + [tex]\frac{\partial}{\partial y}[/tex][tex](y(x^{2} + y^{2} + z^{2})^{\frac{-1}{2}}[/tex] + [tex]\frac{\partial}{\partial z}[/tex][tex](z(x^{2} + y^{2} + z^{2})^{\frac{-1}{2}}[/tex]

Take the partial derivative of x first.

let u = [tex]x^{2} + y^{2} + z^{2}[/tex]

and a = [tex] xu^{2}[/tex]

[tex]\frac{\partial u}{\partial x}[/tex] = 2x

[tex]\frac{\partial a}{\partial u}[/tex] = [tex]\frac{-1}{2}[/tex]x[tex]u^{\frac{-3}{2}}[/tex]

[tex]\frac{\partial a}{\partial x}[/tex] = -[tex]x^{2}[/tex][tex]u^{\frac{-3}{2}}[/tex]

= -[tex]x^{2}[/tex]([tex]\frac{1}{\sqrt{(x^{2} + y^{2} + z^{2})}}[/tex][tex])^{3}[/tex]

The other derivatives would give similar answers, and the final answer would be

-[tex]\frac{x^{2}}{r^{3}}[/tex]-[tex]\frac{y^{2}}{r^{3}}[/tex]-[tex]\frac{z^{2}}{r^{3}}[/tex]

This is apparently the incorrect answer, can anybody help?