- Gradient theorem
The gradient theorem, sometimes also known as the fundamental theorem of calculus for line integrals, says that a
line integralthrough a gradientfield (any irrotational vector fieldcan be expressed as a gradient) can be evaluated by evaluating the original scalar fieldat the endpoints of the curve::
It is a generalisation of the
fundamental theorem of calculusto 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 as potential, 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.
Let be a 0-form (
Let "L" be a 1-segment (
curve) from p to q.
Stokes' theorem::But because ,:
Restricting the curve to Euclidean space and expanding in Cartesian coordinates:
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