Manipulation of calculus notation I'm trying to get my calculus back up to scratch after not using it for 20 odd years. During my research, I've just seen this on https://physics.info/kinematics-calculus/:
$$a = \frac{dv}{dt}$$
$$dv = a\ dt$$
$$\int_{v_0}^v dv = \int_0^{\Delta t} a\ dt$$
Is this a valid manipulation of the calculous notation? My understanding is that the d/dt syntax is just notational, so I'm surprised to see it being treated as if the dt and dv where just algebraic variables that can be manipulated in this way.
Specifically, multiply both side by $dt$ is a surprise. Supposing for example we had this:
$$a = \frac{d}{dt}\ f(x)$$
Now, if I do the same thing as above, I get this:
$$d\ f(x) = a\ dt$$
Which looks like nonsense.

The short answer is no, don't do it and it doesn't make sense. You are safe in treating it is just a heuristic that works to simplify some calculations. Since it isn't a defined  operation (unless the author has defined exactly what it means somewhere else), you can rewrite arguments without using it.
There is a framework, namely differential geometry, where these operations are given meaning, but it involves learning a whole new language - that of differential forms. 
