Editing Work (physics) (section)

===Path independence===
If the work for an applied force is independent of the path, then the work done by the force, by the [[gradient theorem]], defines a potential function which is evaluated at the start and end of the trajectory of the point of application. This means that there is a potential function {{math|''U''('''x''')}}, that can be evaluated at the two points {{math|'''x'''(''t''<sub>1</sub>)}} and {{math|'''x'''(''t''<sub>2</sub>)}} to obtain the work over any trajectory between these two points. It is tradition to define this function with a negative sign so that positive work is a reduction in the potential, that is
<math display="block"> W = \int_C \mathbf{F} \cdot d\mathbf{x} = \int_{\mathbf{x}(t_1)}^{\mathbf{x}(t_2)} \mathbf{F} \cdot d\mathbf{x} = U(\mathbf{x}(t_1))-U(\mathbf{x}(t_2)).</math>

The function {{math|''U''('''x''')}} is called the [[potential energy]] associated with the applied force.  The force derived from such a potential function is said to be [[Conservative force|conservative]]. Examples of forces that have potential energies are gravity and spring forces.

In this case, the [[gradient]] of work yields
<math display="block" qid=Q11402> \nabla W = -\nabla U = -\left(\frac{\partial U}{\partial x}, \frac{\partial U}{\partial y}, \frac{\partial U}{\partial z}\right) = \mathbf{F},</math>
and the force '''F''' is said to be "derivable from a potential."<ref>{{Cite book|last=Taylor|first=John R.|url=https://books.google.com/books?id=P1kCtNr-pJsC&pg=PA117|title=Classical Mechanics|date=2005|publisher=University Science Books|isbn=978-1-891389-22-1|language=en}}</ref>

Because the potential {{mvar|U}} defines a force {{math|'''F'''}} at every point {{math|'''x'''}} in space, the set of forces is called a [[force field (physics)|force field]].  The power applied to a body by a force field is obtained from the gradient of the work, or potential, in the direction of the velocity {{math|'''V'''}} of the body, that is
<math display="block" qid=Q25342>P(t) = -\nabla U \cdot \mathbf{v} = \mathbf{F}\cdot\mathbf{v}.</math>