Math Wiki
Advertisement

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.

Advertisement