Motivation behind notation $\int_C P \, dx + Q \, dy$

Question

I think the notation $$ \int_C P \, dx + Q \, dy $$ is a bit confusing. I understand fairly well the notation $\int_C \vec{F}\cdot d\vec{r}$ and I understand from my question here that they are the same.

My follow up questions are

1. What is the motivation behind using the notation $\int_C P \, dx + Q \, dy$ instead of $\int_C \vec{F} \cdot d\vec{r}$? That is, what is the motivation behind the notation $$ \int_C P \, dx + Q \, dy \text{ ?} $$ Again, the reason I ask is because I think it looks confusing.

2. In a comment to my other question, it was stated that it is best to avoid this notation, is there a reason for this?

I am mostly interested in the answers to the first question, but I welcome all.

Answer 1

1. The integral $\int_C \vec F \cdot d\vec r$ can be evaluated in two different ways. One way you can evaluate this integral is by parameterizing it by some path $\vec v(t)$ over $a\le t\le b$ and then use the relation $$\int_C \vec F \cdot d\vec r = \int_a^b \left[\vec F(\vec v(t))\cdot \vec v'(t)\right]dt \tag{1}$$ to convert this into a Riemann integral. This is probably the method that will make the most sense to you as most multivariable calculus classes only cover Riemann integrals.
   However there is another way we can evaluate this integral: we can multiply out the dot product and obtain the integral of the differential (1-)form $$\int_C \vec F \cdot d\vec r = \int_C Pdx+Qdy \tag{2}$$ Both $(1)$ and $(2)$ are perfectly valid integrals -- but they are different types of integrals.
2. As I'm the one who wrote the comment you're referencing, I guess I should be the one to expand on it. I'm assuming from this question that you've never seen differential forms before. Differential forms are a very valuable tool for solving integrals, especially when you get to integrals on manifolds a little more abstract than $\Bbb R^2$ or $\Bbb R^3$. However if you've never seen them before then you may end up interpreting the integral $\int_C Pdx+Qdy$ incorrectly. One thing that I've seen multiple students mess up on in my time as tutor is that they think to themselves that integrals have the additive property and thus $\int_C Pdx+Qdy = \int_{x_0}^{x_1} Pdx + \int_{y_0}^{y_1} Qdy$. But this is completely wrong. The problem is that they're treating this integral of a differential form as if it were the same as a Riemann integral.
   It is not.
   Thus to avoid mistakes such as this (there are other possible mistakes that come to mind if you try treating differential forms the same as Riemann integrals) I recommend that until and unless you learn about differential forms, if you ever see an integral of the form $\int_C Pdx + Qdy$, that you parametrize the curve and then convert it into an integral of the form $\int_a^b \left[\vec F(\vec v(t))\cdot \vec v'(t)\right]dt$. This is just a Riemann integral and you are hopefully aware of what it means intuitively and how to evaluate it.
   Though some professors (especially physics professors) will claim that the way to manipulate integrals of the form $\int_C Pdx+Qdy$ is obvious, without a bit of formal knowledge of differential forms my recommendation is that you just avoid them.

If you're interested in differential forms -- as I said they are very useful -- then I recommend first taking a look at this short pdf to get the basic idea. Then if you'd like to learn more there are several good books on the subject. One that I found particularly nice is Harold Edwards' Advanced Calculus: A Differential Forms Approach.

Answer 2

Typically you describe a region in terms of $x$ and $y$ not an abstract parametrization. Further, your choice of parametrization of the path can change an easy integration into a hard one. Sure the $\mathbf{F}\cdot d\mathbf{r}$ is nice to look at, but when it comes down to computing it involves parametrizations which are often messy. In the study of differential forms--the mathematical gold-standard for integration on "nice" sets-- the $dx$ and $dy$ coordinates can be very handy, especially for computing using general theorems such as Stokes' theorem. I would say that is the strongest answer to your first question.

For the second the answer is basically because the differential forms approach generalizes, and it is a way to deal with integration independent of choices of parametrizations. Sure, you can turn line integrals into Riemann integrals with parametrizations, but if that ends up asking you to compute

$$\int_a^b e^{t^2}\,dt$$

that doesn't do you a whole lot of good: basic Riemann integration of functions in one-variable has a lot of short-comings, and often-times we can avoid them with the differential forms approach with the many theorems we have on them. It's easier at the inception for students to feel a connection to familiar ground, but history has shown that differential forms is ultimately a more robust environment in which to do integration.

As for your further confusion from the comments: keep in mind an integral has two components: the integrand and the set of integration. As you say there are two "$d$" terms in the $dx$ and $dy$ notation, but the set of integration is a curve, which is not just in terms of $x$ or $y$, but between the two this is everything. It's the same question as to when do we write a vector $v\in\Bbb R^n$ as $v$ and when do we write it as $(v_1,v_2,\ldots, v_n)$ it's a matter of convenience for proving or understanding the problem at-hand.

And finally, as a general comment: we already had fluxion notation for derivatives: we changed that because new notation is often better or more useful than old notation. In this instance differential forms are better than parametrizations. We also have instances where some things can be easier between two notations depending on the context: instances where there's not a clear winner for every situation. We develop new notation to help us along in problem solving, that's notation's purpose and why bad notation can lead to problems in the development of a subject. It's not a bad thing to have multiple notations for an abstract concept, especially when one or the other might be better in a given practical situation.