Conservative Vector Field Potential

[tazzzdo]

Homework Statement

Let $\vec{E}$($\vec{r}$) = $\vec{r}$/r², r = |$\vec{r}$|, $\vec{r}$ = x$\hat{i}$ + y$\hat{j}$ + z$\hat{k}$ be a vector field in ℝ³. Show that $\vec{E}$ is conservative and find its scalar potential.

Homework Equations

All of the above.

The Attempt at a Solution

$\vec{\nabla}$ $\times$ $\vec{E}$ = $\vec{0}$ $\Rightarrow$ $\vec{E}$ is conservative $\Rightarrow$ $\vec{E}$ = $\vec{\nabla}$f, where f is a scalar function.

∂_xf = x/r²
∂_yf = y/r²
∂_zf = z/r²

r² = x² + y² + z²

By implicit differentiation:

2r$\frac{∂r}{∂x}$ = 2x $\Rightarrow$ $\frac{∂r}{∂x}$ = x/r

And as follows:

$\frac{∂r}{∂y}$ = y/r, $\frac{∂r}{∂z}$ = z/r

$\frac{x}{r}$ $\cdot$ $\frac{1}{r}$ = r^-1$\frac{x}{r}$ = r^-1$\frac{∂r}{∂x}$ = $\frac{∂}{∂x}$(log r)

$\Rightarrow$ f = log r


This is an extra credit question in my Vector Calculus class. I think this is the correct solution, but I'm not positive.

 

[Dick]

It looks fine to me.