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.

## Proof

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.