= Potential Function =

A '''potential function''' is a function ''f'' whose [[Calculus/Gradient|gradient]] is a [[Calculus/VectorField|vector field]].

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== Description ==

If a [[Calculus/VectorField|vector field]] is [[Calculus/VectorField#Conservative_Fields|conservative]], then there exists at least one function ''f'' satisfying ''F = ∇f''. This is called a potential function of ''F''.

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== Identification ==



=== Two Dimensions ===

Consider a [[Calculus/VectorField|vector field]] given as ''F = <P(x,y), Q(x,y)>''. The steps for identifying a potential function ''f'' are:

 1. [[Calculus/Integral|Integrate]] ''P'' with respect to ''x''. This produces a function like ''g(x,y) + h(y)'' where ''h(y)'' is unknown, encompasses the constant ''C'', and accounts for variation in all variables that were held constant for the partial integration.
 2. [[Calculus/Derivative|Derive]] ''g(x,y) + h(y)'' with respect to ''y''. This produces a function like ''g,,y,,(x,y) + h'(y)''.
 3. Set ''g,,y,,(x,y) + h'(y)'' equal to ''Q(x,y)'' and solve for ''h'(y)''.
 4. Integrate ''h'(y)'', solving for the actual ''h(y)''.

The potential functions are expressed as ''g(x,y) + h(y) + C''. The infinite possible values of ''C'' lead to infinitely many potential functions.



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