7
$\begingroup$

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 $ds$?
  • 0
    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.2011-12-28
  • 0
    $c'(t)$ is a vector, $|c'(t)|$ is the length of $c'(t)$2011-12-28
  • 4
    $$\sum f(c(t_i))\frac{|c(t_{i+1})-c(t_i)|}{t_{i+1}-t_i}\rightarrow\int_cfds$$2011-12-28
  • 0
    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.2011-12-28
  • 0
    $ds$ 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.2011-12-28

2 Answers 2