What does it mean to integrate a vector function?

Question

What is the meaning of a vector function and what is the geometric interpretation of integrating such a function?

Answer 1

Integration of a vector function is an ambiguous term, it may mean a lot of completely different integrals.

1. integral of a vector function over an independent scalar parameter, $\int \vec{v}\,dt$ is simply 3 integrals for each component. Example: integration of velocity to get position in space.

2. integral of a vector function over a path: $\int \vec{v}\cdot d\vec{r}$. This integrates the contribution of a vector field along a path, for instance, work as integral of force along a path, Faraday's law (induction) and Ampere's law, circulation in hydrodynamics and so on.

3. integral of a vector function over a surface: $\iint \vec{v}\cdot d\vec{S}$. This tells you the flux of the vector field through a surface. For instance, actual flux of liquid for velocity field, magnetic flux (relevant for induction) and so on.

2 and 3 satisfy special relations (for potential field, (2) gives you zero for closed loop, for sourceless field, (3) gives zero for closed surface,...) - check out Stokes and Gauss laws.

There are other less common options... but the essence is, you have to know what your vector is (is it simply a vector variable? is it defined everywhere in space - a field?), and what your expression means. Usually in physics, it's just one of the natural laws in general form when you get from local to global expression.

Answer 2

Vector function in 3d space is used to represent a function, and a function in three variables.

Velocity is the integration of acceleration function which can be represented as a vector function in space. This integration can be done if the functions along x y and z co-ordinates are parameteized.

So integrating the acceleration vector function here gives you again a vector function which is velocity and you can interpret speed from this function.