SkipCaptcha, Automoderated users, edit
3,266
edits
Changes
Jump to navigation
Jump to search
Line 1:
Line 1: − + Line 6:
Line 6: + + + +
Conservative vector field (view source)
Revision as of 03:42, January 20, 2010
, 03:42, January 20, 2010Mention irrotational.
A '''conservative field''' or '''conservative vector field''' has a [[curl]] of zero:
A '''conservative field''' or '''conservative vector field''' (not related to political conservatism) is a field with a [[curl]] of zero:
:<math>\nabla \times \vec V = (\ \ \frac{\partial V_z}{\partial y} - \frac{\partial V_y}{\partial z},\ \ \ \ \frac{\partial V_x}{\partial z} - \frac{\partial V_z}{\partial x},\ \ \ \ \frac{\partial V_y}{\partial x} - \frac{\partial V_x}{\partial y}\ \ ) = 0</math>
:<math>\nabla \times \vec V = (\ \ \frac{\partial V_z}{\partial y} - \frac{\partial V_y}{\partial z},\ \ \ \ \frac{\partial V_x}{\partial z} - \frac{\partial V_z}{\partial x},\ \ \ \ \frac{\partial V_y}{\partial x} - \frac{\partial V_x}{\partial y}\ \ ) = 0</math>
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.
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.
It gets its name from the fact that, if a force field, such as the gravitational or electric field has a curl of zero, the principle of conservation of energy will hold. This follows from the fact that the accumulated force around any closed loop is zero, so no energy is gained or lost.
An older name for such a field is ''irrotational''. This refers to the fact that such a field lacks "vortices" that go around in circles.
[[Category:vector analysis]]
[[Category:vector analysis]]
[[Category:calculus]]
[[Category:calculus]]
[[Category:mathematics]]
[[Category:mathematics]]