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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.