A vector field is defined as a set of vectors associated with each point in space. For instance, the velocity v(r) in a moving liquid (e.g., a whirlpool) constitutes a vector field. By analogy, a scalar field is a set of scalars associated with each point in space. An example of a scalar field is the temperature distribution T(r) in a furnace. Consider a general vector field A(r). Let dl = (dx, dy, dz) be the vector element of line length. Vector line integrals often arise as

 (1.1)

For instance, if A is a force then the line integral is the work done in going from P to Q. As an example, consider the work done in a repulsive, inverse square law, central field F = -r/|r³|. The element of work done is dW = F . dl. Take P = (, 0, 0) and Q = (a, 0, 0). Route 1 is along the x-axis, so

 (1.2)

The second route is, firstly, around a large circle (r = constant) to the point (a, , 0) and then parallel to the y-axis. In the first part no work is done since F is perpendicular to dl. In the second part

 (1.3)

In this case the integral is independent of path (which is just as well!).