Gradient theorem

The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve.

Let ${\displaystyle \varphi :U\subseteq \mathbb {R} ^{n}\to \mathbb {R} }$ and ${\displaystyle \gamma }$ a curve from p to q. Then

It is a generalization of the fundamental theorem of calculus to any curve in a plane or space (generally n-dimensional) rather than just the real line.

The gradient theorem implies that line integrals through gradient fields are path independent. In physics this theorem is one of the ways of defining a conservative force. By placing φ as potential, ∇φ is a conservative field. 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.

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