Editing Potential energy (section)

=== Derivable from a potential ===
In this section the relationship between work and potential energy is presented in more detail. The [[line integral]] that defines work along curve ''C'' takes a special form if the force '''F''' is related to a scalar field ''U''′('''x''') so that
<math display="block"> \mathbf{F}={\nabla U'} = \left ( \frac{\partial U'}{\partial x}, \frac{\partial U'}{\partial y}, \frac{\partial U'}{\partial z} \right ). </math>
This means that the units of ''U''′ must be this case, work along the curve is given by
<math display="block">W = \int_{C} \mathbf{F} \cdot d\mathbf{x}
= \int_{C} \nabla U'\cdot d\mathbf{x},</math>
which can be evaluated using the [[gradient theorem]] to obtain
<math display="block"> W= U'(\mathbf{x}_\text{B}) - U'(\mathbf{x}_\text{A}).</math>
This shows that when forces are derivable from a scalar field, the work of those forces along a curve ''C'' is computed by evaluating the scalar field at the start point A and the end point B of the curve.  This means the work integral does not depend on the path between A and B and is said to be independent of the path.

Potential energy {{math|1=''U'' = −''U''′('''x''')}} is traditionally defined as the negative of this scalar field so that work by the force field decreases potential energy, that is
<math display="block"> W = U(\mathbf{x}_\text{A}) - U(\mathbf{x}_\text{B}).</math>

In this case, the application of the [[del operator]] to the work function yields,
<math display="block"> {\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|author=John Robert Taylor|title=Classical Mechanics|url=https://books.google.com/books?id=P1kCtNr-pJsC&pg=PA117|date=2005|publisher=University Science Books|isbn=978-1-891389-22-1|page=117}}</ref> This also necessarily implies that '''F''' must be a [[conservative vector field]]. The potential ''U'' defines a force '''F''' at every point '''x''' in space, so the set of forces is called a [[force field (physics)|force field]].