On the 'Derivation' of line integrals The line integral wikipedia page https://en.wikipedia.org/wiki/Line_integral#Derivation gives derivations for both the formulas of a line integral over a scalar field and a vector field.
Is it really correct to call these derivations? I.e. isn't it the case that line integrals are inherently definitions? That is to say there is no way to prove or derive these formulas, as we just define certain integrals to themselves be what we refer to as 'line integrals'.
In the derivation on wikipedia, am I correct in my understanding that actually the Riemann sum they are deriving is the definition of the line integrals? (and it just so happens that these Riemann sums can be expressed neatly in other equivalent forms)
Is it the case that line integrals are useful definitions rather than important theorems? I am wondering mostly because this is a recent example that came up that highlights my confusion sometimes over what people refer to as proofs or derivations. Is there not an important distinction between what definitions yield important theories vs what can be deduced from those theories? I am not sure if I am reading too much into it or if I am possibly missing something key so that in fact line integrals come from some more fundamental maths; but I don't see this from the derivations given.
 A: Of course there are various kinds of line integrals. They all involve a "field" $f$ defined in some region $\Omega\subset{\mathbb R}^d$, $\>d\in\{2,3\}$ in many cases, or $\Omega\subset{\mathbb C}$,  and a curve $\gamma\subset\Omega$. The line integral
$$\int_\gamma f\>d?$$
then wants to capture the "total effect" this $f$ has along $\gamma$.
To arrive at the final definition of such an integral one considers partitions of $\gamma$ into tiny "line elements", the field $f$ (considered constant along such a line element), and is arguing geometrically or physically what the total effect of $f$ along this line element should be. At the end of the discussion (via Riemannian sums) one arrives at a certain integral involving $f$ and a parametric representation of $\gamma$. The whole thing is not a proof, but a motivation why one would consider, e.g., the integral
$$\int_\gamma ds:=\int_a^b\bigl|\gamma'(t)\bigr|\>dt $$
an interesting thing. A posteriori then one would actually prove that this integral is in many interesting cases equal to the length of $\gamma$, the latter defined in geometric terms.
For many such integrals there are indeed theorems that say, these integrals are equal to $2$-dimensional integrals of some derivative of the vector field $f$ over a surface, or they are equal to some "potential difference". But this comes after the setup.
To sum it up: In an article about "line integrals"  a half dozen different line integrals are defined, and their definitions motivated, but there are no proofs of mathematical theorems, like the FTC, or Stokes' theorem.
