Let $ a < b $ be real numbers. Let $ F\colon \mathbb R^n\to \mathbb R^n $ be a $ \mathscr C^1 $ vector field on $ \mathbb R^n $, and let $ \gamma\colon \left[a,b\right]\to \mathbb R $ be $ \mathscr C^1 $ curve.
Today I put my physicist shoes and tried to derive in some way the defining formula for the line integral of $ F $ along $ \gamma $, but I got stuck.
First of all, some notation. I denote with the angle brackets the usual scalar product on $ \mathbb R^n $ and with $$ \int_\gamma \langle F,\mathrm d\gamma\rangle = \int_a^b \langle F(\gamma(t)),\dot\gamma(t)\rangle\,\mathrm d t $$ the line integral of $ F $ along $ \gamma $. I call a subdivision of the interval $ \left[a,b\right] $ a finite set $ \sigma = \{a = t_0 < t_1 < \cdots < t_{n-1} < t_n = b\} $ of points $ t_i\in \left[a,b\right] $. I say that $ \sigma $ is tagged if it's also given a finite set $ \{\tau_1,\dots,\tau_n\} $ of $ \tau_i\in \left[a,b\right] $ such that $ t_{i - 1} < \tau_i < t_i $ for any $ i = 1,\dots,n $. I denote with $ \mathcal T $ the directed set of all the tagged subdivisions of $ \left[a,b\right] $.
After fixing a tagged subdivision $ (\{t_i\}_{i = 0}^n,\{\tau_i\}_{i = 1}^n) $ of $ \left[a,b\right] $ I started playing with the sums $$ \sum_{i = 1}^n \langle F(\gamma(\tau_i)),\gamma(t_i) - \gamma(t_{i - 1})\rangle \label{Riemann}\tag{1} $$ noticing first that I can write $$ \langle F(\gamma(\tau_i)),\gamma(t_i) - \gamma(t_{i - 1})\rangle = \Bigl\langle F(\gamma(\tau_i)),\dot\gamma(t_{i - 1}) + \frac{o_{i - 1}(t_i - t_{i - 1})}{t_i - t_{i - 1}}\Bigr\rangle (t_i - t_{i - 1}) $$ for each $ i = 1,\dots,n $, and organizing then sums $ \eqref{Riemann} $ as $$ \sum_{i = 1}^n \langle F(\gamma(\tau_i)),\gamma(t_i) - \gamma(t_{i - 1})\rangle = \sum_{i = 1}^n \langle F(\gamma(\tau_i)),\dot\gamma(\color{red}{t_{i - 1}})\rangle (t_i - t_{i-1}) + \color{blue}{\sum_{i = 1}^n \Bigl\langle F(\gamma(\tau_i)),\frac{o_{i - 1}(t_i - t_{i - 1})}{t_i - t_{i - 1}}\Bigr\rangle (t_i - t_{i-1})} \label{Line}\tag{2} $$ where $ o_{i-1} $ is a vector valued function defined on a neighborhood of $ 0 $ such that $ \lVert o_{i - 1}(\xi)\rVert/{\lvert \xi\rvert} \to 0 $ for $ \xi\to 0 $.
Now, \eqref{Riemann} differ from a Riemann sum for the line integral of $ F $ along $ \gamma $ in two places. There's that red $ \color{red}{t_{i - 1}} $, and there's the blue term in $ \eqref{Line} $. The blue term is probably insignificant: acting like I have the slightest grasp of what I'm doing I just wrote that $$ \lim_{(\{t_i\}_{i = 0}^n,\{\tau_i\}_{i = 1}^n)\in \mathcal T}\sum_{i = 1}^n \Bigl\langle F(\gamma(\tau_i)),\frac{o_{i - 1}(t_i - t_{i - 1})}{t_i - t_{i - 1}}\Bigr\rangle (t_i - t_{i-1}) = 0 $$ (this can be made rigorous with small effort, I think). But what about the argument of $ \dot\gamma $?
It's the first time I'm actually dealing with directed limits and such, and I'm not particularly well versed in $ \epsilon $-$ \delta $ arguments, so I'm asking for your help.