# Multivariate integration of vector fields

## Integration of vector fields

### Line integral of vector fields

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

$$\int_C f(r) ds =\int_a^b f(r(t)).r'(t) dt$$