Change of Variable in velocity calculation

Question

We know from physics that $dv=adt$

1) Why is mathematically correct to write $\frac{dv}{dt}dt=adt$ and so considering infinitesimals as numbers ?

2) I want to calculate the definite integral of dv : $\int_{v_{1}}^{v_{1}}dv$ . Now, since $dv=adt$ , i get that $\int_{v_{1}}^{v_{2}}adt$ . But it's not correct. Why i am wrong and what calculus theorems i have to use in order to correctly calculate the integral ?

Answer 1

Partial answer: I believe the right side integral is with respect to time and not with respect to velocity, so your limits would have to change or you'd have to change variables to express dt in terms of dv. And as the limits are identical at v1 an integral should result in zero.

Answer 2

The integration is correct, however you are 'integrating' over a single point yielding 0. You are not considering infinitesimals as numbers either, you are simply using Leibniz notation.