Line Integral Definition - Rigorous?

Question

Is this definition rigorous to some extent or is there any flaw in this formulation?

Answer 1

It captures the bulk of the spirit of a rigorous derivation, but there is more to the story. However, not much is lost in their description. Because this derivation is found in many textbooks and surely on many websites, I will not give the full argument there but rather outline it and leave it to you to fill in the gaps or find a proof you like.

Let $\mathbf{x}(t): [a,b] \to \mathbb{R}^n$ be a $C^1$ path with non-vanishing first derivative for $t \in [a,b]$, and let $\mathbf{F}: \mathbb{R}^n \to \mathbb{R}^n$ be a vector field. Let $a=t_0<\cdots<t_k<\cdots<t_n=b$ be a partition of $[a,b]$. Consider the sum $$ \sum_{k=1}^n \mathbf{F}(\mathbf{x}(t_k^*)) \cdot \Delta \mathbf{x}_k $$ Now use the definition of the derivative $\mathbf{x}'(t)$ to show that when $\Delta t_k= t_k - t_{k-1} \approx 0$ and for appropriate $t_{k-1} \leq t_k^* \leq t_k$ that $\mathbf{x}'(t_k^*) \approx \Delta \mathbf{x}_k/\Delta t_k$. Then $$ \sum_{k=1}^n \mathbf{F}(\mathbf{x}(t_k^*)) \cdot \Delta \mathbf{x}_k \approx \sum_{k=1}^n \mathbf{F}(\mathbf{x}(t_k^*)) \cdot \mathbf{x}'(t_k^*)\Delta t_k $$ Then show that $$ \lim_{\|\Delta t_k\|\to 0} \sum_{k=1}^n \mathbf{F}(\mathbf{x}(t_k^*)) \cdot \mathbf{x}'(t_k^*)\Delta t_k $$ converges to $$ \int_a^b \mathbf{F}(\mathbf{x}(t)) \cdot \mathbf{x}'(t) \;dt $$ which will what you will define as $\displaystyle\int_C \mathbf{F} \cdot d\mathbf{s}$.

Maybe looking over this outline of a proof, you can see all the essential ideas: breaking the path into smaller pieces, approximating the change over these straight line segments, then totaling them up. The rest is just details: making the $\approx$ and convergences justifiable. If you do try to fill in the proof, I'd recommend carefully keeping track of the mesh of the partition and use the Mean Value Theorems. Note the statement was for one 'smooth' path, but you can modify the proof with a bit more effort to prove this for piecewise smooth curves.