Editing Conservative force (section)

==Mathematical description==
A [[force field (physics)|force field]] ''F'', defined everywhere in space (or within a [[simply-connected]] volume of space), is called a ''conservative force'' or ''[[conservative vector field]]'' if it meets any of these three ''equivalent'' conditions:

# The [[Curl (mathematics)|curl]] of ''F'' is the zero vector: <math display="block">\mathbf{\nabla} \times \mathbf{F} = \mathbf{0}. </math> where in two dimensions this reduces to: <math display="block"> \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} = 0 </math>
# There is zero net [[work (physics)|work]] (''W'') done by the force when moving a particle through a trajectory that starts and ends in the same place: <math display="block">W \equiv \oint_C \mathbf{F} \cdot \mathrm{d}\mathbf r = 0.</math>
# The force can be written as the negative [[gradient]] of a [[potential energy|potential]], <math>\Phi</math>: <math display="block">\mathbf{F} = -\mathbf{\nabla} \Phi. </math>

{{math proof| title = Proof that these three conditions are equivalent when ''F'' is a [[force field (physics)|force field]]
|proof= {{Main|Conservative vector field}}
{{glossary}}
{{term|1 implies 2}}{{defn|
Let ''C'' be any simple closed path (i.e., a path that starts and ends at the same point and has no self-intersections), and consider a surface ''S'' of which ''C'' is the boundary. Then [[Stokes' theorem]] says that
<math display="block"> \int_S \left(\mathbf{\nabla} \times \mathbf{F}\right) \cdot \mathrm{d}\mathbf{a} = \oint_C \mathbf{F} \cdot \mathrm{d}\mathbf{r} </math>
If the curl of '''F''' is zero the left hand side is zero – therefore statement 2 is true.
}}
{{term|2 implies 3}}{{defn|
Assume that statement 2 holds. Let ''c'' be a simple curve from the origin to a point <math>x</math> and define a function
<math display="block">\Phi(x) = -\int_c \mathbf{F} \cdot \mathrm{d}\mathbf{r}.</math>
The fact that this function is well-defined (independent of the choice of ''c'') follows from statement 2. Anyway, from the [[fundamental theorem of calculus]], it follows that <math display="block">\mathbf{F} = -\mathbf{\nabla} \Phi .</math>
So statement 2 implies statement 3 ([[Gradient theorem#Converse of the gradient theorem|see full proof]]).
}}
{{term|3 implies 1}}{{defn|
Finally, assume that the third statement is true. A well-known vector calculus identity states that the curl of the gradient of any function is 0. (See [[Vector calculus identities#Curl of gradient is zero|proof]].) Therefore, if the third statement is true, then the first statement must be true as well.

This shows that statement 1 implies 2, 2 implies 3, and 3 implies 1. Therefore, all three are equivalent, [[Q.E.D.]]

(The equivalence of 1 and 3 is also known as (one aspect of) [[Helmholtz decomposition|Helmholtz's theorem]].)
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The term ''conservative force'' comes from the fact that when a conservative force exists, it conserves mechanical energy. The most familiar conservative forces are [[gravity]], the [[electric force]] (in a time-independent magnetic field, see [[Faraday's law of induction|Faraday's law]]), and [[Hooke's law|spring force]].

Many forces (particularly those that depend on velocity) are not [[force field (physics)|force ''fields'']]. In these cases, the above three conditions are not mathematically equivalent. For example, the [[magnetic force]] satisfies condition 2 (since the work done by a magnetic field on a charged particle is always zero), but does not satisfy condition 3, and condition 1 is not even defined (the force is not a vector field, so one cannot evaluate its curl). Accordingly, some authors classify the magnetic force as conservative,<ref name="srivastava1997mechanics">For example, {{Cite book|title=Mechanics|author=P. K. Srivastava | url=https://books.google.com/books?id=yCw_Hq53ipsC|year=2004|publisher=New Age International Pub. (P) Limited|access-date=2018-11-20 |isbn=9788122411126|page=94}}: "In general, a force which depends explicitly upon the velocity of the particle is not conservative. However, the magnetic force (q'''v'''×'''B''') can be included among conservative forces in the sense that it acts perpendicular to velocity and hence work done is always zero". [https://books.google.com/books?id=yCw_Hq53ipsC Web link]</ref> while others do not.<ref>For example, ''The Magnetic Universe: Geophysical and Astrophysical Dynamo Theory'', Rüdiger and Hollerbach, page 178, [https://books.google.com/books?id=GO1QwZtIYdAC Web link]</ref> The magnetic force is an unusual case; most velocity-dependent forces, such as [[friction]], do not satisfy any of the three conditions, and therefore are unambiguously nonconservative.