The gradient theorem, also known as the fundamental theorem of line integrals, is a theorem which states that a line integral taken over a vector field which is the gradient of a scalar function can be evaluated only by looking at the endpoints of the scalar function. In mathematical terms,
As a corollary, the line integral over is path independent, therefore any closed path over will be equal to zero. Gradient vector fields are also known as conservative.
Let be a differentiable function and be its gradient.
Since the derivative of with respect to will be
by the multivariable chain rule, this expression becomes
by the fundamental theorem of calculus.