A conserved vector field is associated with a function whose gradient equals the vector field, which stems from the properties of conservative forces. The work done by a conservative force on a closed curve is zero, unlike non-conservative forces, due to the Fundamental Theorem of Calculus. This theorem states that if a function is differentiable, the integral of its derivative depends solely on the endpoints, not the path taken. Therefore, when evaluating a closed path, the endpoints coincide, resulting in a net work of zero. This relationship highlights the fundamental nature of conservative forces in physics.
