I have some trouble understanding every component of the line integral formula. Say I have a curve $c : [a,b] \mapsto \mathbb R^n$ and a scalar field $f : \mathbb{R^n} \mapsto \mathbb{R}$.
According to Wikipedia, the integral equation is then:
$$\int_c f \;ds = \int_a^b f(c(t)) |c'(t)| \;dt$$
I understand that $f(c(t))$ is the value of the scalar field on each point on the curve, and that $\int_c ds = \int_c |c'(t)|\;dt$ is the length of the curve.
Things I don't understand:
- What is $|h(x)|$, in general? Does it have any meaning outside the context of arc length?
- Is the result of the line integral the sum of all values of $f$ along the curve?…
- … If yes, why is must we multiply $f$ by $ds$?
Best Answer
Generally if $a=(a_1,\ldots,a_n)\in\mathbb{R}^n$, then $|a|=\sqrt{a_1^2+\cdots+a_n^2}$. If you picture a vector $a$ as an arrow, then $|a|$ is how long the arrow is. When $|c'(t)|$ is equal to a certain number, that means $c(t)$ is moving that number of times as fast as $t$ is changing. So if one intuitively thinks of $dt$ as an infinitely small increment of $t$, then $|c'(t)|\;dt$ is the corresponding infinitely small motion along the curve: distance equals rate times time, and $|c'(t)|$ is the rate and $dt$ is the (infinitely small) time. For that reason $|c'(t)|\;dt$ is identified with $ds$, the infinitely small increment of arc length.
If you multiply $f$ by an infinitely small distance $ds$ moved along the curve, then the integral is the sum of those. In modern and more logically rigorous terms, if you multiply the value of $f$ at a point on the curve by an increment $\Delta s$ of arc length from that point to a nearby point, and add all of those up, and then take the limit as $\Delta s$ approaches $0$, you get the integral. In my view, the logically rigorous definition is justified by its ability to capture the intuition in the older conception, and the older intuitive conception should therefore be remembered.
The formula involving the derivative of $c$ works when $c$ is continuous and piecewise differentiable. Just how far the definition of the line integral, can go when $c$ is not so well behaved is a question that might bear some examination.