Vector field

A vector field is a function that assigns to every point in space a vector. It can be imagined as a collection of arrows, each one attached to a different point in space. For example, the wind (the velocity of air) can be represented by a vector field. This is because in every point one can write an arrow showing the direction of the wind and its strength. 
Vector calculus is the study of vector fields.
Physical examples.
In everyday life, gravity is the force that makes objects fall down. More generally, it is the force that pulls objects towards each other. If we describe for each point in space the direction and strength of the earth's pull, we will get a gravity vector field for the Earth. 
The theory dealing with electricity and magnetism in physics is called electromagnetism. One of the basic assumptions is that there are two vector fields in all space which govern electric and magnetic forces. One is the electric field, which is often written as formula_1 and the second is the magnetic field, which is often written as formula_2.
Atmospheric circulation can be represented by a vector field. The more precisely the vector field describes the actual flow, the better the resulting weather forecast.