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.