What's the point of the fancy notation for surface integrals and line integrals?

Question

Most of the times you see line integrals of a vector field written as this

$$ \int_C\mathbf{F\cdot ds} $$

And surface integrals like

$$\iint_\Sigma \mathbf{F\cdot n}\,\mathrm dS$$

My question is, what's the point of all this symbology? It seems like the $\mathbf {ds}$ and $\mathbf{n}\, \mathrm dS$ are there just to remind the reader "hey! This is a little tiny piece of curve/surface!" or whatever heuristic you have to explain this integrals.

If you know that $C\subseteq \Bbb R^k$ is a curve, and $\Sigma\subseteq \Bbb R^k$ is a surface, why not just write $\int_C \bf F$ and $\iint_\Sigma \bf F$ (or even $\int_\Sigma \bf F$ (although the double integral sign makes more sense, because in the end you end up calculating a double integral))?

Answer 1

Well, it is important to specify which are your infinitesimal variables, as you could integrate your line integral with respect to $x$, $y$, or both. For example, when you use Green's theorem: $$ \oint_C P dx +Qdy =\iint (P,Q) d\vec{S}, $$ it is important to integrate $P$ over $x$ only, and $Q$ over $y$ only.

The same thing applies for surface integrals.

Answer 2

The designation is important. For example, we can write

the scalar $I_1$

$$I_1=\int_S (\hat n\cdot \vec F) \,dS$$

the vector $I_2$

$$I_2=\int_S (\hat n\times \vec F) \,dS$$

and the dyadic (tensor, rank 2) $I_3$

$$I_3=\int_S(\hat n\,\vec F) \,dS$$

The notation for $I$,

$$I=\int_S \vec F\,$$

is ambiguous without explicit designation.

$$$$

Answer 3

It's true that . However, you might note that the same can be said of your ordinary integral on a line: one can easily see if $f$ is a function $[a,b]\rightarrow \mathbb R$, then it's already clear what $$\int_a^bf$$ means, and the $dx$ is just going to be tagging along. However, the advantage of writing $$\int_a^b f(x)\,dx$$ becomes more clear when one has a useful theory of differential forms (i.e. things like $dx$) and can manipulate them meaningfully.

The simplest example is of course doing $u$-substitution where we make a substitution $x=f(u)$ and then correspondingly replace $dx=f'(u)\,du$, which is not an obviously legal thing to do unless we had a $dx$ in the first place. Similarly, in two dimensions, we might rewrite $dA=r\,dr\wedge d\theta$ to move to radial coordinates or make other similar manipulations. Even if it's generally obvious what was intended without this, when it comes to actually evaluating integrals, we might mess around with this last term quite a lot.

Here's one other example which requires a considerable amount of machinery: sometimes one integrates in a more general way where take a set $S$, a function $f$, and a measure $\mu$ (which tells us how "large" subsets of $S$ are) and we integrate as $\int_S f\,d\mu$ where we require $\mu$ to give the set enough structure to define integration. Here, we might have the specification of $\mu$ telling us "are we integrating over a volume or a surface area?" and if we really start taking advantage of it, we can start integrating with respect to masses or charges, which comes in handy in physics (and could not possibly be explicit in the domain).