A Method for Proving Stokes' Theorem

Rather than give the entire proof here (which requires a few more techniques) I'll simply outline the proof. The more or less full version of this proof is given at the end of the chapter on ``Flux through Parameterized Surfaces.''

Stokes' Theorem states that

$$\oint_C \vec{F} \cdot d\vec{r} = \int_S (\vec{\nabla} \times \vec{F}) \cdot \hat{n} dS.$$ (12)

Thus, to prove the theorem, we should show that the two integrals evaluate to the same quantity. To do this, we mimic the proof of Green's Theorem. Break the surface S into many small pieces which are smooth. Now each piece of surface, Delta S_i, we have a corresponding boundary $\Delta C_i$. To compute the circulation, we can simply add up the circulations around each of the $\Delta C_i$'s. The more difficult part is to compute the flux of $\mbox{curl} \vec{F}$ through each of these pieces of surface, $\Delta S_i$.

To to this, we need to find the unit normal vector to the surface. We can get this by parameterizing the surface (see the next section). The parameter curves of the surface then provide a new coordinate system and a way to break up the surface. At one corner of $\Delta S_i$, calculate the tangent vectors to the surface in the directions of each of the parameter curves. Their cross product, when turned into a unit vector, is normal to $\Delta S_i$ at the corner. Assuming that we have broken the surface into small enough pieces that $\mbox{curl} \vec{F}$ is approximately linear on each piece and that each piece of surface is almost a plane, we can now compute the flux of the curl through the surface, taking a linear approximation to $\vec{F}$ on each piece.

To conclude the proof, we need to add up all of the flux integrals and then take the limit as the number of pieces of the surface goes to infinity. This involves a lot of work. The proof in the next section goes through the details for a somewhat less general case where the surface can be described as z = f(x,y). Minor modifications suffice to adjust the proof to the case y = g(x,z) or x = h(y,z). A few more modifications are needed if the surface is only piecewise smooth instead of one of these cases.