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Conservative vector field (view source)
Revision as of 03:28, December 28, 2009
, 03:28, December 28, 2009Quick proof of path invariance.
Its significance is that the line integral of a conservative field, such as a physical force, is independent of the path chosen. In physics, this means that the potential energy (which is determined by a conservative force field) of a particle at a given position is independent of how a particle was moved to its position.
Its significance is that the line integral of a conservative field, such as a physical force, is independent of the path chosen. In physics, this means that the potential energy (which is determined by a conservative force field) of a particle at a given position is independent of how a particle was moved to its position.
The proof of this uses [[Stokes' Theorem]]. Since the curl is zero, any line integral around a closed loop is zero. If there are two paths from point A to point B, the first path from A to B, followed by the second path ''in reverse direction'' from B back to A, constitutes a closed loop, so its line integral is zero. But that's the sum of the first path integral and the negative of the second path integral, so the integrals are equal.
[[Category:vector analysis]]
[[Category:vector analysis]]
[[Category:calculus]]
[[Category:calculus]]
[[Category:mathematics]]
[[Category:mathematics]]