I'm a bit confused with some notation I encounter in physics calculus. Consider this:
Taken from here. Integration operates on functions, correct? What does it mean to integrate $\frac{d{\bf p}}{dt} dt$? Could someone point me to some reading about the manipulation of Leibniz notation?
While in some contexts,
$$ \frac{d\textbf{p}}{dt}dt = d\textbf{p} $$
is correct and is mathematically rigorous, there is a straightforward way to derive impulse as the change in momentum. Consider a function $f(x)$ where $f:\mathbb{R}^m \to \mathbb{R}^n$ and $f \in C^1$. Suppose its derivative is $g = f'(x)$. We can consider integrating $g(x)$ over some interval $I = [x_i, x_f] \subset \mathbb{R}^m$ such that
$$ \int_I g(t)dt = \int_{x_1}^{x_2}g(t)dt = f(x_2) - f(x_1), $$
by the fundamental theorem of calculus. We can apply this result to momentum to get your result. Consider that the momentum $\textbf{p}(t)$ of a particle is simply a function $\textbf{p}: \mathbb{R} \to \mathbb{R}^3$. The impulse is defined as the net external force $\textbf{F}_{ext}(t)$ that a particle experiences over some time interval $I = [t_1, t_2]$, or
$$\textbf{J} \equiv \int_I \textbf{F}(t)dt,$$
where the "ext" has been dropped for brevity. (It is still understood that it represents the net external force.) By definition, $\textbf{F}(t) \equiv \textbf{p}'(t),$ so we can apply the fundamental theorem of calculus to obtain the desired result:
$$\int_{t_1}^{t_2} \textbf{F}(t)dt = \textbf{p}(t_2) - \textbf{p}(t_1) = \Delta \textbf{p}.$$
What does it mean to integrate $\frac{d\mathbf p}{dt}dt$?
First, and in scalar form, recall from elementary calculus that
$$\int_{x_1}^{x_2} dx = x_2 - x_1 $$
Second, recall that
$$f(x + dx) = f(x) + f'(x)dx$$
where
$$f'(x) = \frac{df(x)}{dx} $$
Denoting the differential of $f$ as
$$df = f(x + dx) - f(x)$$
we have
$$df = f'(x)dx$$
Since
$$\int_{x_1}^{x_2} f'(x)dx = f(x_2) - f(x_1) = f_2 - f_1$$
it follows that
$$\int_{f_1}^{f_2} df = f_2 - f_1 = f(x_2) - f(x_1) = \int_{x_1}^{x_2} f'(x)dx$$
Integration operates on functions, correct?
No. Integration dates back to ca. 1670. The notion of a function gradually evolved and didn't get put into its modern form until ca. 1830. Let's look at your expression
$$ \int_{p_1}^{p_2} dp .$$
This is Leibniz's notation, and what he means by it is a sum of infinitely many terms $dp$. Each of these terms is an infinitesimal change in the variable $p$, which doesn't have to be a function of anything.
There is a lot of confusion about the way physicists and engineers use differentials, because there was a period ca. 1880-1960 when it was believed that they were logically suspect and the only correct way to understand calculus was in terms of limits. Non-standard analysis cleared up this confusion and showed that infinitesimals were not logically problematic. A nice book at the freshman calc level that explains the modern way of looking at this is Keisler, https://www.math.wisc.edu/~keisler/calc.html , which is free online.
