» Required Reading » Table of Contents » Chapter 22 22.2 Measures on Surfaces and Stokes Theorem
	Previous Section	Next Section

Suppose now we have two adjacent but not overlapping areas, say rectangles, in a plane. We ask: what entity can we define on the boundaries of each rectangle that will be additive, like the difference of F is on a curve?
What we need is that the contributions to our entity from the common boundary of the two rectangles cancel each other out.

There is an easy way to achieve this: we can integrate something around the outer bounding perimeter of the rectangle, in some standard direction. We always choose the counterclockwise direction as the positive one. If we do so we will integrate up the internal boundary between two adjacent rectangles in one and down that boundary in the other and it will contribute nothing, and we will have our additivity.

The next question to consider is, what kind of entity should we integrate over the boundary of our rectangle (or other region)?

We could in principle integrate either a scalar or vector field in this way; we saw in the last chapter that the line integral of a vector field was the most natural generalization of the ordinary integral to a path integral. The natural additive boundary defined entity for an area is then the integral around its boundary of the line integral of the dot product of some vector v with the path tangent vector:

This entity will be additive on areas A for any vector v for which it is defined, in the plane, or for that matter on any surface that is locally planar, and even on any surface that consists of pieces that are locally planar.
The symbol that we use for an integral of this kind, with a circle on the integral sign, is meant to indicate that we are integrating all around the boundaries of the piecewise locally planar region A.

Why are we doing all this?

The answer becomes apparent if we consider what this integral actually is when the region A is a rectangle with infinitesimal sides. We assume we are working in three dimensional space. We choose coordinates so that A has sides parallel to the x and y axes, and the z axis (or direction of k) is normal to it, and identify its lower left corner as the point (x, y, z).
Then our integral consists of two parts, the vertical and horizontal segments of the integral around A as illustrated here.
The two vertical segments contribute

or

or simply

The integrand here can be identified as and we can write this integral as

By an exactly parallel sequence of steps we can identify the horizontal integrals here as

so that the entire integral becomes

Notice now that we can write the integrand here as and since the z-direction, that of k, is normal to the plane of A, and since dxdy is dA, we can deduce that for this infinitesimal rectangle we have

The fact that both sides of this equation are additive means that we can draw the same conclusion for any region A that we can piece together by putting together small rectangles. This includes regions in a plane, or on any surface in our three dimensional space that can be divided into a finite number of pieces that are each locally planar and rectangular.

With only slightly more effort, we can deduce the same equation for triangular regions A. (When the boundaries of A are slant in the xy plane, the integrals around the outside, having dot products as integrands, break up into two terms, one an integral dx and one dy, and these each have exactly the same forms as occur above, and give rise to the same conclusions.)
We can deduce then that our conclusion holds for any region A on any piecewise locally planar surface with sides that can be well approximated by a polygon.

This is a very important result. It is, as we have seen here, a version of the fundamental theorem of calculus for planar integrals. When applied to integrals over an area in the plane it is called Green's Theorem. When applied to integrals over a general surface it is called Stoke's Theorem. In particular since the integrand on its right side is the curl of the vector v, it is what gives importance and meaning to the concept of curl, as we shall see.