If $F=m\dfrac{dv}{dt}$ why is it incorrect to write $F\,dt=m\,dv$?

Question

My university lecturer told me that:

If $$F=m\dfrac{dv}{dt}$$ it's incorrect to write $$F\,dt=m\,dv\tag{1}$$ but it is okay to write $$\int F\,dt=\int m\,dv$$ for Newtons' second law.

But never explained why $(1)$ is mathematically incorrect.

My high school teacher told me that:

Derivatives with respect to one independent variable can be treated as fractions.

So this implies that $(1)$ is valid.

This is clearly a contradiction as my high school teacher and university lecturer cannot both be correct. Or can they?

Another example of this misuse of derivatives uses the specific heat capacity $c$ which is defined to be $$c=\frac{1}{m}\frac{\delta Q}{dT}\tag{2}$$

Now in the same vain another lecturer wrote that $$\delta Q=mc\,dT$$ by rearranging $(2)$.

Another contraction to the first lecturer. I this really allowed or if it's invalid then which mathematical 'rule' has been violated here?

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EDIT:

In my question here I have used formulae that belong to Physics but these were just simple examples to illustrate the point. My question is much more general and applies to any differential equation in mathematics involving the treatment of derivatives with respect to one independent variable as fractions.

Specifically; Why is it 'strictly' incorrect to rearrange them without taking the integral of both sides?

Answer 1

It is possible that your lecturer is telling you that, on its own, the expression $dt$ is meaningless, whereas $\int...dt$ does mean something quite specific, i.e. an operator or instruction to integrate with respect to $t$.

In contrast, $\delta t$ does mean something specific, i.e. a small increment in the value of $t$.

However, most people are fairly casual about this sort of thing.

Answer 2

More precisely,

$$F=\frac{dp}{dt}=m\frac{dv}{dt}+v\frac{dm}{dt}$$

$$\int F\, dt=\Delta p$$

If $\dfrac{dm}{dt}=0$, then $$\int F\, dt=m\int dv$$

Answer 3

The difference is not so much between university lecturers and highschool teachers as between mathematicians and physicists. Some mathematicians tend to frown on certain procedures that are perfectly acceptable to physicists. I was careful to write "some" because mathematicians familiar with Robinson's framework with infinitesimals do assign a perfectly rigorous meaning to formulas like $F\, dt = m\, dv$; see Keisler's beautiful textbook Elementary Calculus for details.