- Gradient theorem
The

**gradient theorem**, sometimes also known as the**fundamental theorem of calculus for line integrals**, says that aline integral through agradient field (anyirrotational vector field can be expressed as a gradient) can be evaluated by evaluating the originalscalar field at the endpoints of the curve::$phileft(mathbf\{q\}\; ight)-phileft(mathbf\{p\}\; ight)\; =\; int\_L\; ablaphicdot\; dmathbf\{r\}$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 $phi$ as potential, $ablaphi$ 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.**Proof**Let $phi$ be a 0-form (

scalar field ).Let "L" be a 1-segment (

curve ) from**p**to**q**.By

Stokes' theorem ::$int\_\{partial\; L\}\; phi\; =\; int\_L\; dphi$But because $partial\; L\; =\; mathbf\; q\; -\; mathbf\; p$,:$phileft(mathbf\{q\}\; ight)-phileft(mathbf\{p\}\; ight)\; =\; int\_L\; dphi$Restricting the curve to Euclidean space and expanding in Cartesian coordinates:

:$dphi\; =\; sum\_i\; frac\{partial\; phi\}\{partial\; x\_i\}\; dx\_i\; =\; left(frac\{partial\}\{partial\; x\_i\}\; ight)phicdotleft(dx\_i\; ight)\; =\; ablaphicdot\; dmathbf\{r\}$

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