A preview of vector calculus

Section 1.1 A preview of vector calculus

Subsection 1.1.1 Motivation

Have you ever wondered why, in the definition of definite integrals

\begin{equation*} \int_a^b f(x)\ dx\text{,} \end{equation*}

there is always a “\(dx\)” inside the integrand? What does it mean, and why is it there? When you were introduced to definite integrals in Calculus I, they were defined as limits of Riemann sums:

\begin{equation*} \int_a^b f(x)\ dx = \lim_{n \to \infty} \sum_{i=1}^n f(a+i\Delta x) \Delta x, \end{equation*}

where \(\Delta x\) is the width of the rectangles in the Riemann sum. It was then argued that in the limit as \(n \to \infty\text{,}\) the width of the rectangles goes to zero, and somehow \(\Delta x\) becomes “\(dx\)”, which is some sort of infinitesimal width of the rectangles. But what does that mean, really? What is this “\(dx\)” thing inside the definite integral?

And it's not like we can just forget about it, even though many students actually do in first year. :-) We all know how important \(dx\) is: just think of substitution for definite integrals. Without the \(dx\text{,}\) substitution would fail miserably. It must be there. Why?

Things become even more interesting with double and triple integrals:

\begin{equation*} \iint_D f\ dA, \qquad \iiint_D f\ dV, \end{equation*}

with \(dA = dx dy\) and \(dV = dx dy dz\) in Cartesian coordinates. What are these objects \(dA\) and \(dV\text{,}\) and why must they be there in the integrand?

What we will do in this course is provide an answer to this question. Our goal is to define a unified theory of integration for curves, surfaces, and volumes, which will make the appearance of \(dx\text{,}\) \(dA\text{,}\) and \(dV\) natural. To achieve this, we must define a new type of objects that will play the role of integrands: those are called differential forms, or \(n\)-forms. More specifically, one-forms are objects that can be integrated over curves, two-forms over surfaces, and three-forms over volumes. In other words, it was all a big lie: what you should be integrating is not functions, but rather differential forms!

But before we start, we can already identify two key guiding principles for the construction, using what we already know about integration.

We want our theory to be “reparametrization-invariant”. Consider a definite integral in one dimension. Instead of writing \(\int_a^b f(x)\ dx\text{,}\) we would like to write something like \(\int_C \omega\text{,}\) where \(C\) stands for a curve, and \(\omega\) for the integrand, which will be called a "one-form". (We will also allow \(C\) to be a curve in \(\mathbb{R}^2\) and \(\mathbb{R}^3\)). More precisely, to make sense of this expression, we will need \(C\) to be a parametric curve. However, we want \(\int_C \omega\) to be defined intrinsically in terms of the geometry of the curve itself: we do not want the integral to depend on the choice of parametrization. This is key. This constraint will be satisfied if the integrand \(\omega\) transforms in a specific way under reparametrizations of the curve; in one dimension this will reproduce the substitution formula for definite integrals.
We want our theory to be “oriented”. Consider \(\int_C \omega\) as above. As mentioned, to make sense of this expression we will work with a parametric curve \(C\text{.}\) But once we parametrize a curve, we introduce a choice of “orientation”: we select one of the two endpoints as the starting point, and we introduce a “direction of travel along the curve” (the orientation is given by the direction of the tangent vector, or velocity, for a parametric curve). If we do a reparametrization of the curve that reverses the orientation, should the integral remain invariant? The answer is no! We see this directly in basic calculus: we know that \(\int_b^a f(x)\ dx = - \int_a^b f(x)\ dx\text{.}\) If we interpret the first integral as being over the interval \([a,b]\) with direction of travel from \(a\) to \(b\text{,}\) and the second integral as being over the same interval but with the reverse orientation, then we see that exchanging the direction of travel over the interval \([a,b]\) changes the sign of the integral.

Putting these two properties together, we see that if \(C\) and \(C'\) are two parametrizations of the same curve that preserve the orientation, we must have \(\int_C \omega = \int_{C'} \omega\text{,}\) while if they reverse the orientation, we must have \(\int_C \omega = -\int_{C'} \omega\text{.}\) This is what it means to say that the theory should be “oriented” and “reparametrization-invariant”.

This brief exposition focused on curves, but following these two guiding principles we will develop a unified theory of oriented integrals not only over curves, but over surfaces and volumes as well, using the machinery of differential forms. Along the way we will discover the beautiful intricacies of vector calculus, culminating with the very important Stokes' Theorem. Let us embark on this journey together!

Subsection 1.1.2 Vector calculus and differential forms: two sides of the same coin

What we will study in this course is known as vector calculus. There are two main approaches to vector calculus. On the one hand, there is the “traditional” approach, which involves concepts such as gradient, curl, div, etc. It relies heavily on the geometry of \(\mathbb{R}^3\text{,}\) and is very explicit. But at first it seems like a complicated amalgation of strange constructions and definitions that satisfy all kinds of intricate identities. My recollections of learning vector calculus is that it involves many formulae that appear to come out of nowhere and that one needs to learn by heart. Not fun.

On the other hand, there is the “modern” approach, pioneered by Cartan, which relies on the definition of differential forms. This approach is a little more abstract, but is much more unified and elegant. It brings together all the concepts of vector calculus in a unified formalism, from which all the identities and formulae come out naturally. It also does not rely on the geometry of \(\mathbb{R}^3\text{,}\) and is naturally generalized to \(\mathbb{R}^n\) (even though we will focus on \(\mathbb{R}^3\) in this course). I remember this feeling of “ah, now this all makes sense!” when I learned differential forms later on in my studies.

In this course we will take the perhaps non-traditional approach of introducing vector calculus directly through the unified formalism of differential forms, guided by the exposition of the previous subsection. The challenge is to make the concepts accessible to second-year students, stripping them down from the fancier definitions of differential geometry. But I truly believe that this is possible, and that it will make the whole theory of vector calculus much more interesting and unified, and less reliant on brute force memorization.

But, at the same time, it is important for students to be fluent with the traditional concepts of vector calculus. Indeed, students who will study topics like fluid mechanics, electromagnetism, applied mathematics, etc. will repeadtedly encounter vector calculus, usually expressed in the traditional language. Moreover, traditional concepts such as grad, div, curl, are often useful for explicit calculations. So in this course we will translate all concepts from differential forms to standard vector calculus every step of the way.

In the end, the goal is for students to be fluent with both approaches: to see the beauty and elegance of differential forms, and to be able to use the traditional approach for explicit calculations.