We may wish to integrate along a curve in a vector field.
We previously showed that we can write a curve as a function on the real line:
\(r:[a,b]\rightarrow C\)
The integral is therefore the sum of the function at all points, with some weighting. We write this:
\(\int_C f(r) ds=\lim_{\Delta s rightarrow 0 }\sum_{i=0}^n f(r(t_i))\Delta s_i\)
In a vector field we use
\(\int_C f(r) ds =\int_a^b f(r(t)).r'(t) dt\)