Undergraduate Vector Calculus Notation Mess

Question

Question 1: What are your arguments in favor of the big array of different notations used in the context of undergraduate vector calculus: line integrals, surface integrals (of scalars and fields), Stoke's and divergence theorems, etc.

Because I personally believe that, even from the first introduction of the students to multivariable integration, we should abandon all that mess with creatures that look like $$ {\oint\!\!\oint}_S \bar{F} \, dS \, $$ and $$ \iiint_{V } (\nabla \cdot \mathbf{F}) \, dV = \oint\!\! \oint_{\partial V} \mathbf{F} \cdot d \mathbf{S} $$ in favor of replacements like $$ \int_S \vec{F} \, $$ and $$ \int _V \nabla \cdot F = \int_{\partial V} F \ . $$ A few points in support of my viewpoint:

1) What do the $dr$, $ds$, and $\cdot \vec{ds}$, in case of line integrals, and $dS$, $d\sigma$, $\cdot \vec{dS}$, in case of surface integrals, really mean? They discourage the question of whether the integrals are intrinsic, because, for instance, $dr$ finds meaning as $|r'(t)| \, dt$ only after a parameterization of the curve. So, is $dr$ meaningful as an intrinsic "length element"? Is $dS$ intrinsic? (Adding notation does not answer the real question of the nature of such integrations.)

2) What is that stupid loop over integrals $\oint$? Just to say that the curve/surface is closed?! One would know the curve/surface they are integrating over is closed anyway. Such integrals are always preceded by assumptions beforehand that give the context: Let $D$ be a closed region in $\mathbb{R}^3$ and $f$... Let $S \subset \mathbb{R}^3$ be a smooth oriented surface and $\bar{F}$ a vector field... So, even the simplest notation will unambiguously tell us what is being integrated over what.

3) If we use successive $\int$ signs because the domain is 2 or 3 dimensional then are you going to keep adding new $\int$ for $n$-dimension? See Wiki's Vector Calculus article for $$ \underbrace{ \idotsint_{V \subset \mathbb R^n} }_{n} (\nabla \cdot \mathbf{F}) \, dV ={\underbrace{ \oint \!\cdots\! \oint_{\partial V}}_{n-1}} \mathbf{F} \cdot d \mathbf{S} $$ Furthermore, multiple integral signs might be confused with iterated integrals as in Fubini's theorem. Yes, by the same reasoning, I like $$ \int_{[0,1]^2} f \quad \text{or} \quad \int_{[0,1]^2} f \, dx\,dy $$ over $$\iint_{[0,1]^2} f \, dx\,dy . $$ Don't even ask about $\int_0^1\int_0^1 f \, dx\,dy$.

Question 2: I know later on many people switch to the shorter notation, but isn't it about time to save our new generations the pain of having to learn a ton of nonsense notation conventions by saying goodbye to some "notation and terminology [that] was established by Gibbs and Edwin Bidwell Wilson in their 1901 book, Vector Analysis"?! (Wiki, same article.)

Question 3: Are there pedagogical benefits to sticking to the old notation?

Maybe we should throw the words "Hausdorff measures" into the air, just like we use "area/surface element", and agree that $ d\mathcal{H}^1 $ is for integration over 1-D objects, curves, straight or bent, $d\mathcal{H}^2$ is for integration domains that are 2D, (curved) sheets in space, etc. Then for instance $$ \int_C f(x) \, d\mathcal{H}^1 (x) \quad \text{and} \quad \int_S f(x) \, d\mathcal{H}^2(x). $$ Then every time you look at an integral you will immediately see the dimension of the object your are integrating over?!

If you can talk from experience teaching Calculus to undergraduates please do so.

Answer 1

This is a service course for students who are mostly engineering majors. Therefore any drastic change in notation like this is likely to be a bad idea. Leaving out $d\textbf{S}$ and $dV$ would be particularly unfortunate, since leaving out the $dx$ is such a common student mistake anyway in freshman calculus.

Also, any notation that has the wrong units is a non-starter. As a typical application, we have Gauss's law for electric fields,

$$\int \textbf{E}\cdot d\textbf{S}=4\pi k q,$$

where $k$ is the Coulomb constant and $q$ is the enclosed charge. The SI units on the left are

$$\frac{\text{N}}{\text{C}}\cdot\text{m}^2,$$

and these match the units on the right. Without the $d\textbf{S}$, the units don't match. Your students' physics and engineering instructors are pounding it into them over and over that they need to check their units, and that if an equation is wrong in terms of dimensional analysis, then it's guaranteed to be a mistake. Neither the students nor these instructors are going to be happy if you start undoing that careful training.

The other things you mention, however, are more trivial differences in notation, and not particularly standardized anyway in the physics and engineering literature. Nobody is going to care whether you write $\int$ or $\oint$ , and whether you write $\iint$ or $\int$.

I know later on many people switch to the shorter notation, but isn't it about time to save our new generations the pain of having to learn a ton of nonsense notation conventions [...]

This hinges on your opinion that these are "nonsense." Getting the notation to correctly show the units is certainly not "nonsense," although many mathematicians might not be aware of it.

Are there pedagogical benefits to sticking to the old notation?

If you're proposing a radical change in notation, then the burden of proof is on you to show that there are pedagogical benefits to changing the notation, and that these benefits are huge.

Maybe we should tell them about Hausdorff measures first

Please be realistic. At my school, almost none of the students who take this course are math majors. About half of them have serious deficits in basic algebra and arithmetic, such as not being able to simplify the expression $2/(3/4)$.

Answer 2

I think you are being harsh in your criticism of the classical notation. Of course, at the mathematician's end of the spectrum, the notation you promote towards the end of your question has merit. But I have taught vector calculus for many years, and find the classical notations that provoke you do in fact help learners decode theorems and calculations. These classical notations tell stories that capture the interest of calculus students, and make connections between the words in theorems and symbolism.

Students who take vector calculus in the USA are from a mixture of majors, such as physics, chemistry, engineering, and mathematics. Some are double majors. Many take a course in electrodynamics from the physics department, and need to see connections with vector calculus taught in the mathematics curriculum.

In physics, they might see Maxwell's equations in integral form, while in vector calculus, they see the theorems of Gauss and Stokes. Maintaining reasonably consistent notation between the two courses is important. And I don't think my physics colleagues would take warmly to a suggestion that they reform their notation to one that mathematicians recommend.

Mathematicians do not have a body that oversees notations, such as the Académie Française, which oversees matters related to the French language. A top-down movement to reform the notation of vector calculus could never get a foothold. It would only happen by random, incremental evolution. Aspiration to such a reform reminds me of the aspiration of an international language such as Esperanto.

An important quality of mathematics is its ubiquity with applications. We should not aspire to burn the bridge between vector calculus and classical electromagnetism taught by our colleagues in physics departments.