The fundamental theorem for line integrals states that if a line integral passes through a gradient field, it can be evaluated by evaluating the endpoints of the curve in the initial scalar field. If a smooth curve C exists and a function with a gradient vector is continuous on curve C, then the line integral can be calculated using the fundamental theorem of calculus for single integrals.
In order for the theorem to work, the line integral must be independent of the path because all that is required are the initial and final points of curve C. The gradient theorem for line integrals can only work in a conservative vector field because it is path-independent. The use of the gradient theorem simplifies the process of finding line integrals, since all that is required is to find out if the vector field is conservative.
The easiest way to determine if a vector field is conservative is by finding its potential function. An important definition relating to line integrals is a closed path, which refers to paths with endpoints that are the same, such as circles. Another important definition is a simple path, which is a path that does not cross itself, such as a circle.