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| A '''potential function''' is a function ''f'' whose [[Calculus/Gradients|gradient]] is a [[Calculus/VectorField|vector field]]. | A '''potential function''' is a function ''f'' whose [[Calculus/Gradient|gradient]] is a [[Calculus/VectorField|vector field]]. |
Potential Function
A potential function is a function f whose gradient is a vector field.
Description
If a vector field is conservative, then there exists at least one function f satisfying F = ∇f. This is called a potential function of F.
Identification
Two Dimensions
Consider a vector field given as F = <P(x,y), Q(x,y)>. The steps for identifying a potential function f are:
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.
Derive g(x,y) + h(y) with respect to y. This produces a function like gy(x,y) + h'(y).
Set gy(x,y) + h'(y) equal to Q(x,y) and solve for h'(y).
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.