散度、旋度与麦克斯韦方程组

The two identities

The Laplacian

Maxwell's equations as a reading exercise

The inverse-square field's ledger

Which statement is guaranteed by the identities, for any smooth $V$ and $\mathbf{F}$?

Maxwell's $\nabla \cdot \mathbf{B} = 0$ combined with the divergence theorem says what about magnetic flux out of any closed surface?

The field $q\,\hat{\mathbf{r}}/|\mathbf{r}|^2$ has divergence $0$ away from the origin. Why does a sphere around the origin still catch flux $4\pi q$?

For $V = x^2 y + z$, give the $z$-component of $\nabla \times \nabla V$, as a plain number.

Give the Laplacian of $V = x^2 + y^2 + 3z^2$, as a plain number.