Stokes' Theorem

We are now ready for the generalized Green's Theorem, known as Stokes' Theorem. If we have circulation density of $\vec{F}$ in a region and we want the total circulation, we should integrate the density over the region. Thus,

If the following conditions hold:

1.

S is a smooth oriented surface,

2.

C is the oriented boundary of S with orientation determined from that of S by the right-hand rule, and

3.

$\vec{F}$ is a vector field with $\nabla \times \vec{F}$ defined at every point of S,

then we can say that

$$\oint_C \vec{F} \cdot d\vec{r} = \int_S \vec{\nabla} \times \vec{F} \cdot \hat{n} dS.$$

where $\hat{n}$ is the unit normal vector to S at each point.

Stokes' Theorem simply says that the total circulation of $\vec{F}$ around C is the same as the integral of the maximum circulation density over any surface S with C as its boundary. Thus, we can choose S to simplify the calculations as needed.

It is also worth noting that if we let $\vec{F} = F_1 \hat{i} + F_2 \hat{j}$ then Stokes's Theorem directly reduces to Green's Theorem in the plane.