Rombu is the hawt.

The gradient theorem, sometimes also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field (any irrotational vector field can be expressed as a gradient) can be evaluated by evaluating the original scalar field at the endpoints of the curve: This article is about path integrals in the general mathematical sense, and not the path integral formulation of physics which was studied by Richard Feynman. ... Horizontal line (use sparingly)d grade for the grade or gradient of roads and other geographic features. ... In vector calculus, an irrotational or conservative vector field is a vector field whose curl is zero. ... In mathematics and physics, a scalar field associates a scalar to every point in space. ...

It is a generalisation of the fundamental theorem of calculus to any curve on a line rather than just the real line. The fundamental theorem of calculus is the statement that the two central operations of calculus, differentiation and integration, are inverse functions of one another. ...

The gradient theorem implies that line integrals through irrotational vector fields are path independent. In physics this theorem is one of the ways of defining a "conservative" force. By placing φ as potential, is a conservative force. Work done by conservative forces does not depend on the path followed by the object, but only the end points, as the above equation shows

Proof

Let φ be a 0-form (scalar field). A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics and physics, a scalar field associates a scalar to every point in space. ...

Let L be a 1-segment (curve) from p to q. In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ...

By Stokes' theorem Stokes theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. ...

But because ,

Restricting the curve to Euclidean space and expanding in Cartesian coordinates: