Gradient theorem

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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:

$\phi\left(\mathbf{q}\right)-\phi\left(\mathbf{p}\right) = \int_L \nabla\phi\cdot d\mathbf{r}$

It is a generalisation of the fundamental theorem of calculus to any curve on a line rather than just the real line.

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, $\nabla\phi$ 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.

[edit] Proof

Let $φ$ be a 0-form (scalar field).

Let L be a 1-segment (curve) from p to q.

By Stokes' theorem:

$\int_{\partial L} \phi = \int_L d\phi$

But because $\partial L = \mathbf q - \mathbf p$ ,

$\phi\left(\mathbf{q}\right)-\phi\left(\mathbf{p}\right) = \int_L d\phi$

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

$d\phi = \sum_i \frac{\partial \phi}{\partial x_i} dx_i = \left(\frac{\partial}{\partial x_i}\right)\phi\cdot\left(dx_i\right) = \nabla\phi\cdot d\mathbf{r}$