Force field (physics)

In physics, a force field is a vector field that describes a non-contact force acting on a particle at various positions in space. Specifically, a force field is a vector field formula_1, where formula_2 is the force that a particle would feel if it were at the point formula_3.
Work.
Work is dependent on the displacement as well as the force acting on an object. As a particle moves through a force field along a path "C", the work done by the force is a line integral
This value is independent of the velocity/momentum that the particle travels along the path. 
Conservative force field.
For a conservative force field, it is also independent of the path itself, depending only on the starting and ending points. Therefore, the work for an object travelling in a closed path is zero, since its starting and ending points are the same:
If the field is conservative, the work done can be more easily evaluated by realizing that a conservative vector field can be written as the gradient of some scalar potential function:
The work done is then simply the difference in the value of this potential in the starting and end points of the path. If these points are given by "x" = "a" and "x" = "b", respectively: