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

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

Double integral of vector fields

Surface integral for vector fields

Stoke’s theorem

The divergence theorem

Stoke’s theorem