In this chapter we will define what we mean by a smooth curve in the plane and what is meant by its arc length. These definitions are a good bit more tricky than one might imagine. Indeed, it is the subtlety of the definition of arc length that prevented us from defining the trigonometric functions in terms of wrapping the real line around the circle, a definition frequently used in high school trigonometry courses. Having made a proper definition of arc length, we will then be able to establish the formula C=2πr for the circumference of a circle of radius r.

By the “plane,” we will mean R²≡C, and we will on occasion want to carefully distinguish between these two notions of the plane, i.e., two real variables x and y as opposed to one complex variable z=x+iy. In various instances, for clarity, we will use notations like x+iy and (x,y), remembering that both of these represent the same point in the plane. As x+iy, it is a single complex number, while as (x,y) we may think of it as a vector in R² having a magnitude and, if nonzero, a direction.

We also will define in this chapter three different kinds of integrals over such curves. The first kind, called “integration with respect to arc length,” will be completely analogous to the integral defined in Section 6.1 for functions on a closed and bounded interval, and it will only deal with functions whose domain is the set consisting of the points on the curve. The second kind of integral, called a “contour integral,” is similar to the first one, but it emphasizes in a critical way that we are integrating a complex-valued function over a curve in the complex plane C and not simply over a subset of R². The applications of contour integrals is usually to functions whose domains are open subsets of the plane that contain the curve as a proper subset, i.e., whose domains are larger than just the curve. The third kind of integral over a curve, called a “line integral,” is conceptually very different from the first two. In fact, we won't be integrating functions at all but rather a new notion that we call “differential forms.” This is actually the beginnings of the subject called differential geometry, whose intricacies and power are much more evident in higher dimensions than 2.

The main points of this chapter include:

Our first project is to make a satisfactory definition of a smooth curve in the plane, for there is a good bit of subtlety to such a definition. In fact, the material in this chapter is all surprisingly tricky, and the proofs are good solid analytical arguments, with lots of ϵ's and references to earlier theorems.

Whatever definition we adopt for a curve, we certainly want straight lines, circles, and other natural geometric objects to be covered by our definition. Our intuition is that a curve in the plane should be a “1-dimensional” subset, whatever that may mean. At this point, we have no definition of the dimension of a general set, so this is probably not the way to think about curves. On the other hand, from the point of view of a physicist, we might well define a curve as the trajectory followed by a particle moving in the plane, whatever that may be. As it happens, we do have some notion of how to describe mathematically the trajectory of a moving particle. We suppose that a particle moving in the plane proceeds in a continuous manner relative to time. That is, the position of the particle at time t is given by a continuous function f(t)=x(t)+iy(t)≡(x(t),y(t)), as t ranges from time a to time b. A good first guess at a definition of a curve joining two points z₁ and z₂ might well be that it is the range C of a continuous function f that is defined on some closed bounded interval [a,b]. This would be a curve that joins the two points z₁=f(a) and z₂=f(b) in the plane. Unfortunately, this is also not a satisfactory definition of a curve, because of the following surprising and bizarre mathematical example, first discovered by Guiseppe Peano in 1890.

Be careful to realize that we're talking about the “range” of f and not its graph. The graph of a real-valued function could never be the entire square. This Peano function is a complex-valued function of a real variable. Anyway, whatever definition we settle on for a curve, we do not want the entire unit square to be a curve, so this first attempt at a definition is obviously not going to work.

Let's go back to the particle tracing out a trajectory. The physicist would probably agree that the particle should have a continuously varying velocity at all times, or at nearly all times, i.e., the function f should be continuously differentiable. Recall that the velocity of the particle is defined to be the rate of change of the position of the particle, and that's just the derivative f^' of f. We might also assume that the particle is never at rest as it traces out the curve, i.e., the derivative f^'(t) is never 0. As a final simplification, we could suppose that the curve never crosses itself, i.e., the particle is never at the same position more than once during the time interval from t=a to t=b. In fact, these considerations inspire the formal definition of a curve that we will adopt below.

Recall that a function f that is continuous on a closed interval [a,b] and continuously differentiable on the open interval (a,b) is called a smooth function on [a,b]. And, if there exists a partition of [a,b] such that f is smooth on each subinterval then f is called piecewise smooth on [a,b]. Although the derivative of a smooth function is only defined and continuous on the open interval (a,b), and hence possibly is unbounded, it follows from part (d) of Exercise 22. that this derivative is improperly-integrable on that open interval. We recall also that just because a function is improperly-integrable on an open interval, its absolute value may not be improperly-integrable. Before giving the formal definition of a smooth curve, which apparently will be related to smooth or piecewise smooth functions, it is prudent to present an approximation theorem about smooth functions. Exercise 20. asserts that every continuous function on a closed bounded interval is the uniform limit of a sequence of step functions. We give next a similar, but stronger, result about smooth functions. It asserts that a smooth function can be approximated “almost uniformly” by piecewise linear functions.

Theorem 6.1.

Let f be a smooth function on a closed and bounded interval [a,b], and assume that |f^'| is improperly-integrable on the open interval (a,b). Given an ϵ>0, there exists a piecewise linear function p for which

1.  |f(x)–p(x)|<ϵ for all x∈[a,b].

2.  

That is, the functions f and p are close everywhere, and their derivatives are close on average in the sense that the integral of the absolute value of the difference of the derivatives is small.

Proof

Because f is continuous on the compact set [a,b], it is uniformly continuous. Hence, let δ>0 be such that if x,y∈[a,b], and |x–y|<δ, then |f(x)–f(y)|<ϵ/2.

Because |f^'| is improperly-integrable on the open interval (a,b), we may use part (b) of Exercise 22. to find a δ^'>0, which may also be chosen to be <δ, such that ∫_a^a+δ'|f^'|+∫^b_b–δ^'|f^'|<ϵ/2, and we fix such a δ^'.

Now, because f^' is uniformly continuous on the compact set there exists an α>0 such that |f^'(x)–f^'(y)|<ϵ/4(b–a) if x and y belong to and |x–y|<α. Choose a partition of [a,b] such that x₀=a,x₁=a+δ^',x_n–1=b–δ^',x_n=b, and x_i–x_i–1<min(δ,α) for 2≤i≤n–1. Define p to be the piecewise linear function on [a,b] whose graph is the polygonal line joining the n+1 points for 1≤i≤n–1, and That is, p is constant on the outer subintervals and determined by the partition, and its graph between x₁ and x_n–1 is the polygonal line joining the points For example, for 2≤i≤n–1, the function p has the form

(6.1)

on the interval So, p(x) lies between the numbers and for all i. Therefore,

(6.2)

Since this inequality holds for all i, part (1) is proved.

Next, for 2≤i≤n–1, and for each we have which, by the Mean Value Theorem, is equal to for some So, for each such we have and this is less than ϵ/4(b–a), because |x–y_i|<α. On the two outer intervals, p(x) is a constant, so that p^'(x)=0. Hence,

(6.3)

The proof is now complete.

Now we are ready to define what a smooth curve is.