${\bf F}=m{\bf a}$ versus ${\bf F}=\frac{\mathrm d{\bf p}}{\mathrm dt}$ alternate expressions (differential) I am currently collecting research about theories to do with accelerated motion, and scientists who were involved in the development of such theories.
Intrinsically, Newton's Second Law came up. While this has most of time appeared in the form of $F=ma$, Wikipedia gave two forms, first in terms of differential terms: 
$$F=\frac{\mathrm dp}{\mathrm dt}\text{ and then }F=m\frac{\mathrm dv}{\mathrm dt.}$$
I was wondering if these different forms change the meaning of the equation somehow, as I have read that the differential form is "more accurate". However, mathematically, $\mathrm dv/\mathrm dt$ is the same as acceleration and hence $F=m\,\mathrm dv/\mathrm dt$ should be the same as $F=ma$ (I think).
Could someone explain if these different forms actually make a difference?
 A: Newton's second law is only valid for constant-mass systems, so
$$
{\bf F}=\frac{\rm d {\bf p}}{{\rm d}t}\equiv\frac{\rm d}{{\rm d}t}\left(m{\bf v}\right)\equiv m\frac{\rm d {\bf v}}{{\rm d}t}\equiv m{\bf a}
$$ 
That is, they are all equivalent definitions for Newtonian mechanics.
A: As you state in your question, they are exactly the same: take into account that $p = m·v$, and therefore, $$F = \frac{dp}{dt} = \frac{d(m·v)}{dt} = m· \frac{dv}{dt} = m·a$$
, as m should be independent of $t$ to use the second Newton law.
There is no difference between the different forms you show in the question, but the differential forms are always more accurate than those that are not so: if you want to compute the acceleration of a car, you could either compute its velocity at instants 0s and 10s or do the same between 0 and 1s. The first number would be more like an average for those first ten seconds and the second an average for the first second. Therefore, if you want to know the acceleration of the car at the instant 0s, probably the second approach is far better than the first. The differential form assumes that the $\Delta t $ is practically 0, so the answer is going to be the best.
A: Newton's law is $F=\frac{dp}{dt}$. This is more complete than $$F=ma$$ or the equivalent: $$F=m\frac{dv}{dt}$$ because it allows $m$ to vary with time, not just $v$. $F=ma$ is an approximation that assumes $m$ is constant. To see this, consider: $$F=\frac{dp}{dt}=\frac{d(mv)}{dt}=v\frac{dm}{dt}+\frac{m·dv}{dt}$$
where we have used $p=mv$. We can see that if we assume that the $dm/dt$ on the RHS is $0$, then we have $$\frac{dp}{dt}=m\frac{dv}{dt}$$
To answer your question about formatting: to get displayed equations, put them in double $ signs. I find this reference helpful.
