Examples of Stokes' Theorem

Example 1. Evaluate the circulation of $\vec{F}$ around the curve C where C is the circle x² + y² = 4 that lies in the plane z= -3, oriented counterclockwise with $\vec{F} = y\hat{i} + xz^3\hat{j} - zy^3\hat{k}$.

Take as the surface S in Stokes' Theorem the disk $x^2 + y^2 \le 4$ in the plane z = -3. Then $\hat{n} = \hat{k}$ everywhere on S. Further, $\vec{\nabla} \times \vec{F} = -3z(y^2 + xz)\hat{i} + (z^3 - 1)\hat{k}$ so

$$\oint_C \vec{F} \cdot d\vec{r} = \int_S \vec{\nabla} \times ... ...\cdot \hat{n} dS = \int_S (z^3 - 1) dS = \int_S ((-3)^3 - 1) dS$$

$$= -28 \int_S dS = -28 (\mbox{area of S}) = -28(4\pi) = -112\pi.$$

Example 2. Find the work done by the force $\vec{G} = 2xy^3 \sin z \hat{i} + 3x^2 y^2 \sin z \hat{j} + x^2y^3 \cos z\hat{k}$ in the displacement around the curve of the intersection of the paraboloid z = x² + y² and the cylinder (x-1)² + y² = 1.

Notice that $\vec{G}$ is a conservative vector field since $\vec{\nabla} \times \vec{G} = \vec{0}$. Thus, by Stokes' Theorem, the work done around any closed curve, and this one in particular, is zero, since work is simply a line integral.