# Conservative vector field

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A conservative field or conservative vector field (not related to political conservatism) is a field with a curl of zero:

$\nabla \times \vec V = (\ \ \frac{\partial V_z}{\partial y} - \frac{\partial V_y}{\partial z},\ \ \ \ \frac{\partial V_x}{\partial z} - \frac{\partial V_z}{\partial x},\ \ \ \ \frac{\partial V_y}{\partial x} - \frac{\partial V_x}{\partial y}\ \ ) = 0$

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.

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.

## Irrotational and Solenoidal Vector Fields

An older name for a conservative field is irrotational. This refers to the fact that such a field lacks "vortices" that go around in circles.

An older name for a field with a divergence of zero is solenoidal. This refers to the fact that, according to Maxwell's Equations, the divergence of the magnetic field is zero, and "solenoid" is a traditional term for an electromagnet.

Both of these terms seem to have fallen out of use around 1960. [1][2][3]

## References

1. Page, Leigh, Introduction to Theoretical Physics, Van Nostrand, 1952
2. Constant, F.W., Theoretical Physics, Addison-Wesley, 1958
3. Schwartz, M., S. Green, W.A.Rutledge, Vector Analysis, Harper, 1960