0
$\begingroup$

I want to apply Stokes' theorem in a problem whose geometry dictates that the "curl $\vec{F} \cdot \vec{n}$" integrand of the surface integral must equal one. Thus curl $\vec{F}$ must equal $\vec{n}$, and since my surface is the sphere, $\vec{n}$ is the radial vector on the surface of the sphere. the last step before I convert this surface integral to a path integral (using Stokes' theorem) is to define the vector field $\vec{F}$ whose curl is the radial vector $\vec{r}$. I'm rusty at div, grad, curl and all that, and probably overlooking something obvious. I know that the radial vector is the polar angle unit vector $\hat{\theta}$ cross the azimuthal unit vector $\hat{\phi}$, but I need the vector field $\vec{F}$ whose curl is $\vec{r}$, not the two vectors whose cross product is $\vec{r}$. I tried directly integration the expression for curl $\vec{F} = \vec{r}$ to guess what $\vec{F}$ must be and I got close but no cigar. Any help appreciated...

$\endgroup$
1
$\begingroup$

There is no vector field $\bf F$ that satisfies $$\nabla \times {\bf F} = {\bf \hat{r}} ,$$ where $\bf \hat r$ denotes the unit radial vector field on $\Bbb R^3 - \{ 0 \}$:

If there were, taking the divergence would give $$\nabla \cdot {\bf r} = \nabla \cdot (\nabla \times {\bf F}) = 0 .$$

On the other hand, the usual formula for divergence in spherical coordinates gives $$\nabla \cdot {\bf r} = \frac{2}{r} \neq 0 ,$$ where $r$ is the radial spherical coordinate.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.