## Approaching Green’s Theorem via Riemann Sums

- JENNIE BUSKIN, SCOTT A. TAYLOR
- 2013

- Published 2013

The basic theorems of vector calculus are illuminated when we replace the original 3 stooges of vector calculus, Grad, Div, and Curl, with combinatorial substitutes. Grad, Div, and Curl are the three stooges of Vector Calculus: their loveable, but hapless, interactions are viewed with mixtures of delight, puzzlement, and bewilderment by thousands of students viewers. The classic episodes involving Grad and Curl include: Episode 1 Horsing Around: In this hilarious episode, we see that a vector field running in circles does no work if and only if it is a gradient field. Episode 2 Violent is the Word for Curl: Nothing happens when Curl hits Grad who hits a scalar field. Episode 3 Grips, Grunts, and Green’s: Mr. Green tries to circulate around a boundary only to find Curl appearing in surprising places. Despite the delight that often greets these classic episodes2, audiences often have some difficulty in making sense of the basic plot lines. We maintain that these issues are due, in part, to Grad, Div, and Curl’s excellent acting. With great aplomb they manage to combine ideas from calculus, geometry, and topology. In this viewer’s guide, we show how the essence of each episode is clarified if we substitute the coarse actors3 Tilt, Ebb, and Whirl for the good actors Grad, Div, and Curl in each of the classic episodes. (Although, as we mentioned, we leave the episodes involving Ebb to the true aficianados.) These coarse actors merely approximate the good actors; they are defined without the use of limits. 1Fans of the 3 stooges of Vector Calculus will recognize that Div has been neglected in our list of classic episodes. Throughout this viewer’s guide, we invite the reader to fill in the missing episodes which feature this neglected character. 2See Section 1 for precise statements of the episodes/theorems 3“What are the outstanding characteristics of a Coarse Actor? Firstly I should say a desperate desire to impress. The true Coarse Actor is most anxious to succeed. Of course, he is hampered by an inability to act or to move, and a refusal to learn his lines, but no one is more despairing if he fails. In reality, though, Coarse Actors will never admit that they have done badly. Law One of Coarse Drama states: ‘In retrospect all performances are a success”’ [G, page 29] 1 3 STOOGES OF VECTOR CALCULUS 2 Section 1 describes the scenery, introduces the actors, and shows how the classic episodes can be reinterpreted so as to make the basic plot lines more evident. In addition to their simplicity, the episodes with the coarse actors have another advantage over the classic episodes: they do not have to consider the possibility that paths intersect infinitely many times. For example, letting h(t) = { t2 sin(1/t) t 6= 0 0 t = 0 for t ∈ [0,1], the curves γ(t) = (t,0) and ψ(t) = (t,h(t)) are distinct smooth curves that intersect infinitely often in a neighborhood of the origin. Such curves create challenges that are usually ignored in vector calculus classes4. In the knock-off episodes, however, any two simple curves that intersect infinitely many times actually coincide. Not only do the knock-off episodes add conceptual clarity, they can also be used to reconstruct the original episodes. Section 2 shows that, in fact, the only drawback to the knock-off episodes is their scenery and that by refining the scenery, the acting improves. In more conventional language: by taking limits we can recover the classic episodes from our approximations. In particular, we show that the coarse actor Whirl becomes the good actor Curl and our imitation Episode 3 becomes the actual Episode 3. Finally, Section 3, shows how the Good Actor’s Guild (also known as de Rham cohomology5) to which Grad, Div, and Curl belong has a lot in common with the Coarse Actor’s Guild (also known as simplicial cohomology) to which Tilt, Ebb, and Whirl belong. These labor unions organize the actors and clarify their working environment. As always, there’s fine print: To simplify matters, we work for the most part in 2-dimensions (although in Section 3 we point to resources for generalizing these ideas to higher dimensions). Also, there are various topological issues (usually pertaining to the classification of surfaces and the Schönflies theorem) that are ignored. The cognoscenti can fill in the missing details without problems, while the new viewer won’t notice their absence. 1. THE SCENERY AND ACTORS We begin by reviewing the scenery and actors from the classic episodes and then we construct the cheap scenery and introduce the coarse actors. 4See for example [C, page 441] 5Strictly speaking, the cohomology theory given by Grad, Div, and Curl shouldn’t be called de Rham cohomology since de Rham cohomology is usually defined using differential forms. However, the two cohomology theories are similar enough that we appropriate the name. 3 STOOGES OF VECTOR CALCULUS 3 1.1. The classic scenery. The action takes place on a compact, orientable, smooth surface S embedded in Rn. Smooth means that there is a tangent plane at every point x of S and the tangent planes vary continuously as x moves around the surface. Figure 1 shows a smooth surface in R3. FIGURE 1. A smooth surface in R3. It is the image of the function Φ(s, t) = (s, t,st) for (s, t) ∈ [−1,1]× [−1,1]. Most of the surfaces that appear in this viewer’s guide will be embedded in R2. At each point of such a surface, the tangent plane coincides with R2 itself. 1.1.1. Fields. A scalar field on S is a function f : S→ R that is differentiable and whose derivative is continuous. (A differentiable function with continuous derivative is said to be of class C1). A vector field6 on S is a C1 function F : S→ R2. When n = 2 (i.e. when S is embedded in the plane), we often write F = ( M N ) where M and N are C1 functions from S to R. Generally, a scalar field f on a surface is pictured by shading the surface by making points with large f values light and points with small f values dark. A vector field F on a surface is pictured by drawing an arrow based at x ∈ S pointing in the direction of F(x) and of length ||F(x)||. We think of a vector field as telling us the direction of motion and the speed of motion. Figure 2 shows a scalar field and a vector field on the unit disc. A path on S is a continuous function φ : [a,b]→ S for some interval [a,b] ⊂ R. Unless we say otherwise, we require the path to be piecewise C1. That is, there are t0, . . . , tm ∈ [a,b] with a = t0 < t1 < .. . < tm−1 < tm = b 6Vector fields should really take values in the tangent plane to the surface at the points, but we stick with the traditional vector calculus definition. 3 STOOGES OF VECTOR CALCULUS 4 FIGURE 2. On the left is the scalar field f (x,y) = x2 + y2 on the unit disc in R2 and on the right is (a portion of) the vector field F(x,y) = (−y,x)/ √ x2 + y2 on the unit disc. such that the restriction of φ to each subinterval [ti, ti+1] is continuously differentiable with non-zero derivative at every point. (At the endpoints of the interval we require that the one-sided derivatives exist, are continuous, and are non-zero.) If φ and ψ are two paths with the same image in S, we say that they have the same orientation if, as t increases φ and ψ traverse the image of each C1 portion of the image in the same direction. Otherwise, we say that φ and ψ have opposite orientations. We indicate the orientation by drawing arrows on the image of the path as in Figure 3. The path φ is a closed curve if φ(a) = φ(b) and the path φ is embedded if it is one-to-one on the intervals [a,b) and (b,a]. FIGURE 3. The image of the path φ(t) = (.5cos(3t), .5sin(t)) for 0 ≤ t ≤ 2π in the unit disc. It is a non-embedded closed curve. The orientation of the path is marked with an arrow. 1.1.2. Integrals. The integral of a scalar field f over a path φ : [a,b]→ S is defined to be ∫

@inproceedings{BUSKIN2013The3S,
title={The 3 Stooges of Vector Calculus and Their Impersonators: a Viewer’s Guide to the Classic Episodes},
author={JENNIE BUSKIN and PHILIP PROSAPIO and Scott A. Taylor},
year={2013}
}