Curl: Local Circulation, Coordinate Formulas, and Applications

All flashcards

What are common notations for the curl of a vector field $F$?

$\operatorname{curl} F$, $\nabla \times F$, and, in some texts, $\operatorname{rot} F$.

How is a component of curl defined using a shrinking loop?

For unit normal $\hat{u}$ at $p$, it is $\lim_{|A|\to 0}\frac{1}{|A|}\oint_C F\cdot d\mathbf{r}$, where $C$ bounds the oriented patch.

Which coordinate plane corresponds to each Cartesian component of curl?

The $x$-, $y$-, and $z$-components correspond to circulation in the $yz$-, $zx$-, and $xy$-planes, respectively.

What is the product rule for the curl of a scalar times a vector field?

$\nabla\times(\phi F)=\nabla\phi\times F+\phi(\nabla\times F)$.

How can a vector potential change without changing its curl?

Replacing $W$ with $W+\nabla f$ leaves $\nabla\times W$ unchanged.

Quiz questions

1. For an oriented surface $S$ with boundary $\partial S$, which equation states the Kelvin–Stokes theorem?Answer: $\int_S (\nabla\times F)\cdot\hat{n}\,dS=\oint_{\partial S}F\cdot d\mathbf{r}$
2. For $F=(F_x,F_y,F_z)$ in Cartesian coordinates, what is the $z$-component of $\nabla\times F$?Answer: $\frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y}$
3. For a sufficiently smooth vector field $F$, what is $\nabla\cdot(\nabla\times F)$?Answer: $0$
4. Why do curl formulas in orthogonal curvilinear coordinates include scale factors?Answer: They account for the physical lengths associated with coordinate changes.
5. For a sufficiently smooth vector field $F$, which identity gives its curl of curl?Answer: $\nabla\times(\nabla\times F)=\nabla(\nabla\cdot F)-\nabla^2F$