Potential Function

Topics: Vector Field

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(definition)

Given $F:{\mathbb{R}}^n\to V_n$ a vector field, we say that $f:{\mathbb{R}}^n\to \mathbb{R}$ is a potential function of $F$ if:

$$\nabla f=F$$

…where $\nabla f$ denotes the gradient of $f$.

Example

Let $F(x,y)=⟨x,y⟩$.

Is $f:{\mathbb{R}}^2\to \mathbb{R}$, $f(x,y)=\frac{x^2}{2}+\frac{y^2}{2}$ a potential function of $F$?

Let’s obtain its gradient:

$$\nabla f=⟨\frac{\partial f}{\partial x},\frac{\partial f}{\partial y}⟩=⟨x,y⟩=F$$

As we can see, $\nabla f=F$, so $f$ is effectively a potential function of $F$.

When a vector field has a potential function, we say that it is conservative.