The integrand of the double integral on the left is the flux, through the surface $S$, of the curl of F. On the right, the tangential component of F is integrated around the bounding curve $C$, yielding the circulation of F around $C$. Stokes' theorem balances the fields net vorticity (circulation) flowing through the surface $S$, and the average circulation of F around the bounding curve $C$. Vorticity (or twist, rotation) of F on the surface $S$ is measured by the curl of F. Integrating the normal component of the curl of F on the surface $S$ allows local swirls that oppose each other to cancel out, leaving just the uncanceled parts on the boundary curve $C$ to reckon with. This residual vorticity is along the bounding curve $C$ and accumulates along $C$ in the line integral on the right side of the Stokes' formula.

Example 9.9.1

Apply Stokes' theorem to the vector field $\mathbf{F}\=z \mathbf{i}-x \mathbf{j}-y \mathbf{k}$; the curve $C$, the unit circle with center at the origin; and the upper hemisphere as the capping surface $S$.

Example 9.9.2

Show that the flux of the curl of  $\mathbf{F}\=z \mathbf{i}-x \mathbf{j}-y \mathbf{k}$ through the unit sphere centered at the origin is necessarily zero.

Example 9.9.3

Apply Stokes' theorem to the vector field $\mathbf{F}\=y z \mathbf{i}\+x^{2}z \mathbf{j}\+x y \mathbf{k}$; the curve $C$, the unit circle with center at the origin; and a closed cylinder with lid in the plane $z\=a\>0$ as the capping surface $S$.

Example 9.9.4

Apply Stokes' theorem to the vector field $\mathbf{F}\=x y \mathbf{i}\+y z \mathbf{j}\+x z \mathbf{k}$; the curve $C$, the triangle with vertices $\left(0\,0\,20\right)\,\left(30\,0\,0\right)\,\left(0\,24\,0\right)$; and the first-octant portion of the plane $4 x\+5 y\+6 z\=120$ as the capping surface $S$.

Example 9.9.5

Apply Stokes' theorem to the vector field $\mathbf{F}\=x y \mathbf{i}\+y z \mathbf{j}\+x z \mathbf{k}$; the curve $C$, the triangle with vertices $\left(0\,0\,0\right)\,\left(12\,0\,0\right)\,\left(0\,8\,0\right)$; and the first-octant portion of the plane $2 x\+3 y\+4 z\=24$ along with the coordinate planes $x\=y\=0$ as the capping surface $S$. Hint: $S$ has three sides but is open at the bottom.

Example 9.9.6

Apply Stokes' theorem to the vector field $\mathbf{F}\=x^2 y \mathbf{i}\+y z \mathbf{j}\+x z \mathbf{k}$; the curve $C$, the ellipse $x^2\+4 y^2\=1$; and the upper half of the ellipsoid $x^2\+4 y^2\+6 z^2\=1$ as the capping surface $S$.

Example 9.9.7

Show that the flux of the curl of $\mathbf{F}\=x^2 y \mathbf{i}\+y z \mathbf{j}\+x z \mathbf{k}$ through $S$, the upper portion of the ellipsoid $x^2\+4 y^2\+6 z^2\=1$ closed with the interior and boundary of an ellipse in the plane $z\=0$, is necessarily zero.