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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 $ds$?
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An integral is not a sum, as it takes into account the value of $f$ at infinitely many points, but sort of analogous to a sum. –  Alex Becker Dec 28 '11 at 19:45
    
$c'(t)$ is a vector, $|c'(t)|$ is the length of $c'(t)$ –  yoyo Dec 28 '11 at 20:10
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$$\sum f(c(t_i))\frac{|c(t_{i+1})-c(t_i)|}{t_{i+1}-t_i}\rightarrow\int_cfds$$ –  yoyo Dec 28 '11 at 20:17
    
The line integral is the weighted average of $f$ over the curve times the length of the curve, where the average is taken such that equal-length portions of the curve receive equal weight. –  mjqxxxx Dec 28 '11 at 20:22
    
$ds$ is just a notation, whenever you write $d$ in front of something in an integral, it denotes we are integrating a function with respect to that certain object or certain measure. –  Shuhao Cao Dec 28 '11 at 21:25
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2 Answers

The place where I finally understood integration in several dimensions was Bachmann's brilliant book A Geometric Approach to Differential Forms, in particular pages 27-33. (this expands on yoyo's comments)

Basically, the term |c'(x)| comes about when you actually try to compute what the integral should be. The idea is that whatever our notion of integral, for reasonably behaved functions and domains it should be approximated by Riemann sums (If I remember correctly, which I might not, the insistance of the domains to be submanifolds and the functions to have continuous derivatives makes the Lebesgue and Riemann integral coincide, so we can comfortably use Riemann sums).

Your Riemann sum is of course going to look like $\sum f(c(t_i))l(c(t_{i-1}),c(t_i))$ where $l(c(t_{i-1}),c(t_i))$ is the length of the curve between $c(t_{i-1})$ and $c(t_i)$. For simplicity, let $x_i=c(t_i)$. Computing the length of the curve is hard, but we can approximate it by the linear distance between the points, i.e. we should have that $l(c(t_{i-1}),c(t_i))~|c(t_i)-c(t_{i-1})|$, so that the Riemann sums approximating the integral ought to look like $\sum f(c(t_i))|c(t_i)-c(t_{i-1})|$.

These sums, however, live in the $n$-dimensional space that where the curve lives in. If we want to reduce them to a single-variable integral on the domain, we need something that looks like $\sum g(t_i)(t_i-t_{i-1})$. Since $c'(t_i)(t_i-t_{i-1})~c(t_i)-c(t_{i-1})$ by definition, we see that all we need to do is set $g=f\circ c \cdot |c'|$, which would make the two Riemann sums $\sum f(c(t_i))|c(t_i)-c(t_{i-1})|$ and $\sum f(c(t_i))|c'(t_i)|(t_i-t_{i-1})$ agree in the limit (this should be checked; it's a good exercise to test one's understanding of convergence properties). On the left-hand side you get what should be the integral over the curve $\int _c f$, and on the right-hand side you obtain the integral $\int_0^1f\circ c\cdot |c'|$.

The moral of the story is that in order to integrate (in this way) we need the extra information on how to measure the linear differences $c(t_i)-c(t_{i-1}$. The linear differences, however, are essentially vectors (they live in $\mathbb R^n$) so we need a function that takes in a vector and spits out a real number. Furthermore, in order for the summations to agree in the limit, it is necessary that the measuring function commutes with positive scalars, that is, $\phi(\lambda v)=\lambda\phi(v)$ for $\lambda>0$, and is continuous. If these two properties are satisfied, then in the limit the function $\lambda$ becomes a function on the tangent space, that is, on the tangent vectors $c'(t)$. An example of such a function is the $|\cdot|$ function, which is specified by $ds$.

In general if $\phi$ transforms correctly under change of coordinates (either multiplied by $|\det|$ or $\det$ of a transformation), one obtains what are known as densities or differential forms respectively (the latter being linear functions on the vectors and thus nicer to work with). So to answer your three questions in this context:

  • $|h(x)|$ is actually a function on the tangent space which when you compute an integral you have to plug-in the derivative (tangent vector) $c'$
  • The result of the integral depends on the density or differential form you've chosen to integrate against. If integrating against ds, then you can interpret the result as the "sum" of the values of $f$, but if integrating with respect to say, $dx$ or $dx+dy$, it would be the "signed sum" of the values of $f$, depending on the direction of the curve (e.g. the sign of the integral changes if you change the direction of the curve).
  • We multiply by $ds$ because that is what allows us to translate the integral from living in the domain of $\mathbb R^n$ to living in the 1d domain of the curve. Why that is the correct thing to do can be understood from trying to approximate the integrals with Riemann sums.

I recommend Bachman for a much better exposition of the underlying ideas.

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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.

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