The fundamental theorems in two and three dimensions are among the most important and influential results of all time in any subject. They provide the keys that allow representation of the laws governing electric and magnetic phenomena as differential equations, which is the source of our understanding of the structure of physics and to the development of radio and all that has followed from it.
Instead of simply stating these fundamental theorems we begin by exploring the question: what natural forms can they take?
The integrals we have defined: on the real line, in the complex plane, on a path in several dimension, or on surfaces or volumes have the important property that they are finitely additive. By this we mean that if you define any one of them on two non-overlapping pieces their value is the sum of what it is on either piece.
Certainly area under a curve in an interval obeys this property; and in fact, it is the basis for all of the generalizations of this concept that we have defined.
In every generalization remember, we took an appropriate definition of a "piece": (which was in turn, interval on a curve in the complex plane, or on a path in several dimensions, or on a small element of area in the plane or on a surface or an element of volume in three dimensions.)
Then multiplied an appropriate "measure" of its size by some function f defined on it; and defined this product on a large piece to be the sum of its values on a partition of that piece (into "Riemann sums") of its value on smaller pieces.
We then took limits as the maximum diameter of the pieces approached zero.
In each case we could claim that for a continuous function f the resulting sum
converged to a limiting value which we called an integral.
Now we ask the question, are there other natural entities, like these that have this same "additive" property, value on the "sum" of several pieces is the sum of its value on each piece.
We do this because we want to relate appropriate integrals to other things, and it is clear that any other thing that might be equal to an integral of some kind in general, must share at least this property.
The easiest answer to this question determines the structure of fundamental theorem in each context.
To begin we address this question for one dimensional integrals where we have already seen the answer.
We ask, what construct other than area under a curve defined by a function f has this additive property when defined on intervals of the real line?
There is a simple and natural answer: if we have an interval [a, b] on the real line, a natural additive entity is F(b) - F(a), the difference between the function F at the endpoints of the interval, or equivalently, the change in F between a and b, is additive.
Obviously, the sum of the changes in F over two completely disjoint intervals is the sum over both; more interestingly, the change of F over two adjacent intervals is the sum of its changes over each. We have
(F(b) - F(c)) + (F(c) - F(a)) = F(b) - F(a)
which in words says that the contribution from the intermediate point c, which is the front end of one interval and the back end of another, cancels out and we have only contributions from the ends of the resulting large interval.
This answer is not very exciting, and its one dimensional generalizations to paths in the complex plane or to paths in Euclidean space are equally straightforward.
On a path C in the complex plane the difference between F (here now F is a real or complex valued function) at the endpoints of C is finitely additive.
If we divide a path into two sub-paths, or concatenate one path to the end of another, the difference between the values of F at the ends of the resulting sub-paths sum to the difference over the entire path according to the equation above without any modification.
Again, given a scalar field F defined in more than one dimensions, and given a path C in it, the same additive property holds on C for the change in F.
In fact we used this additivity in the Chapter 21 in developing the fundamental theorem of calculus in these contexts.
The only interesting issue, then, is what analogues to differences on a path can we find when we are considering an area in a plane, or on a surface, or a volume, instead of an interval on a path? We are particularly interested in entities which involve the boundaries of the given region which is what F(b) - F(a) represents for an interval on a path.