

6.4 Why is it Important Whether a Vector Field is a Gradient?


We will soon define integrals of vector fields along paths. When a field v
is a gradient, such integrals depend only on the endpoints of integration and
not on the actual path chosen to get from one to the other. Thus integrating
it over a closed path (with no endpoints) gives 0. This simplifies evaluation
of such integrals considerably. We shall see this soon. (The integral of the
gradient of around
a closed path can give any multiple of 2
depending on how many times the path winds around the origin.)
Also, it is often considerably simpler to describe the field in terms of the
function it is a gradient of, than by expressing it directly.