The idea of Stokes' theorem

Stokes' theorem will combine the ideas we developed for Green's theorem with what you just learned about the curl of a vector field. I recommend that you glance back at those pages to put this discussion in context.

Recall that Green's theorem states that, given a two-dimensional vector field F, the integral of the "microscopic circulation" of F over the region D inside a closed curve C is equal to the total circulation of F around C, as suggested by the equation

$$\int_{{C}}^{} \iint_{D}^{}$$

and depicted in the figure below.

(If this figure doesn't make sense, consult the page on Green's theorem.)

Recall that Green's theorem applies only to two-dimensional vector fields and to regions in the two-dimensional plane. Stokes' theorem generalizes Green's theorem to three dimensions. For starters, let's take our above picture and simply embed it in three dimensions. Then, our curve C becomes a curve in the xy-plane, and our region D becomes a surface S in the xy-plane whose boundary is the curve C (if C is the boundary of S, we often write C = $\partial$S).

The next question is what the microscopic circulation along a surface should be. For Green's theorem, we found that

(where k is the unit-vector in the z-direction). We wanted the component of the curl in the k direction because this corresponded to microscopic circulation in the xy-plane. Similarly, for a surface, we will want the microscopic circulation along the surace. This corresponds to the component of the curl that is perpendicular to the surface, i.e,

where n is a unit normal vector to the surface. You can see this using the right-hand rule. If you point the thumb of your right hand perpendicular to a surface, your fingers will curl in a direction corresponding to circulation parallel to the surface.

So, to go from Green's theorem to Stoke's theorem, we've made two changes. First, we've changed the line integral living in two dimensions (Green's theorem) to a line integral living in three dimensions (Stokes' theorem). Second, we changed the double integral of curl F ^. k over a region D in the plane (Green's theorem) to a surface integral of curl F ^. n over a surface floating in space (Stokes' theorem). The required relationship between the curve C and the surface S (Stokes' theorem) is identical to the relationship between the curve C and the region D (Green's theorem): C = $\partial$S.

We summarize by writing out Stokes' theorem as:

$$\int_{{C}}^{} \iint_{{S}}^{} \iint_{{S}}^{}$$

(Recall that a surface integral of a vector field is the integral of the component of the vector field perpendicular to the surface.) We see that the right integral is the surface integral of the vector field curl F. Stokes theorem says the surface integral of curl F over a surface S (i.e., $\iint_{{S}}^{}$curl F ^. dS) is the circulation of F around the boundary of the surface C = $\partial$S (i.e., $\int_{{C}}^{}$F ^. ds).

Once we have Stokes' theorem, we can see that the surface integral of curl F is a special integral. Notice that the integral cannot change if we can change the surface S to any surface as long as the boundary of S is still the curve C It cannot change because it still must be equal to $\int_{{C}}^{}$F ^. ds, which doesn't change if we don't change C. (You could say that the surface integral is "surface independent," but we don't usually use that term.)

For example, staring with a planar surface such as sketched above, we see that surface S doesn't have to be the flat surface inside C. We can bend and stretch S, and the above formula is still true. In the below CVT, you can move the blue point on the slider to change the surface S. For any of those surfaces, the integral of curl F over that surface will be the total circulation of F over the curve C (shown in red). The important thing is that the boundary of the surface S is still the curve C.

Stokes' theorem allows us to do even more. We don't have to leave the curve C sitting in the xy-plane. We can twist and turn C as well. If S is a surface whose boundary is C, it is still true that

$$\int_{{C}}^{} \iint_{{S}}^{}$$

For example, in the below CVT, you can move the blue point on the slider to change the surface S, as before. But, you can also move the green point on the slider to change the curve C (the surface S also changes as you change C, since its boundary has to be C). (Although I didn't draw the green circles to represent the "microscopic circulation," you can imagine what they would look like.)

Note that moving the blue dot on the slider does not change the value of either integral in the above formulas. Since the curve C does not change, the left line integral doesn't change, which means the value of the right surface integral cannot change. On the other hand, moving the green dot on the slider does change the values of the integrals since the curve C changes. The important point is that, even in this case, the left line integral and the right surface integral are always equal.

Orienting the surface properly

We do need to be careful about the orientation of the surface (which is specified by choosing a normal vector n). Remember, changing the orientation of the surface changes the sign of the surface integral. If we choose the wrong n (i.e., the wrong orientation), we could be off by a minus sign.

Look at the first CVT, above, where the "microscopic circulation" is sketched by green circles on the surface. Notice how the arrows on the little green circles (indicating the "microscopic circulation") are aligned with the red arrow indicating the direction of the curve C. If, for example, the arrows on the green circles were going the other direction, the green circles and the red curve wouldn't match and we'd be off by a minus sign.

Looking from the positive z-axis, both the green circles and the red curve indicate counterclockwise circulation. To define the orientation for Green's theorem, this was sufficient. We simply insisted that you orient the curve C in the counterclockwise fashion. For Stokes' theorem, we cannot just say "counterclockwise," since the orientation that is counterclockwise depends on the direction from which you are looking. If you take the first CVT and rotate it 180^o so that you are looking at it from the negative z-axis, the same curve would look like it was oriented in the clockwise fashion. Since the green circles would also look like they are oriented in a clockwise fashion, you can still see that the green circles and the red curve match.

Remember, too, that the curve C can be floating or twisted in any direction. It doesn't have to look as simple as in the above examples. Thankfully, choosing the correct orientation doesn't have to be too difficult if you remember the right hand rule. If you look at your right hand from the side of your thumb, your fingers curl in the counterclockwise direction. Think of your thumb as the normal vector n of a surface. If your thumb points to the positive side of the surface (i.e., the side with the normal vector n), your fingers indicate the circulation corresponding to curl F ^. n. If you put your hand near the edge of the surface, the curve C must be oriented to go around the same direction your fingers are pointing. If the relationship between the normal vector n and the orientation of C doesn't match the relationship between the thumb and fingers of your right hand, you'll be off by a minus sign.

Another way of thinking about the proper orientation is the following. Imagine that you are walking on the positive side of the surface (the side with the normal vector). If you walk near the edge of the surface in the direction corresponding to the orientation of C, then surface must be to your left and the edge C must be to your right.

When the curve C and the surface S are oriented as described above so that Stokes' theorem applies, we say that C is a positively oriented boundary of S.

You can read some examples here.