31. Conservative Vector Fields

We've seen a few classes of vector fields - unit fields, radial, irrotational, incompressible - but there is one special class of fields that is especially useful.

A vector field $\large \vec{F}$ is conservative if $\large \vec{\nabla}\times\vec{F}=\vec{0}$. The name comes from the fact that conservative force fields are linked to conservation of energy in a way we can investigate in class.

Potential Functions

All conservative vector fields are the gradient of some multivariable function $\large V(x,y,z)$, called a potential function. That is, if $\large \vec{F}$ is conservative, then

$$\large \vec{F}(x,y,z) = \nabla V(x,y,z)$$

for some potential function $\large V$ (it is common to use $\large f$, $\large \phi$, or $\large V$ as the notation for a potential function). This could be considered an alternate definition of a conservative field. That is, $\large \vec{\nabla}\times\vec{F}=\vec{0}$ if and only if there is a multivariable potential function $\large V(x,y,z)$ such that $\large \vec{F}(x,y,z) = \nabla V(x,y,z)$.

Note: physicists usually use the sign convention $\large \vec{F}(x,y,z) = -\nabla V(x,y,z)$ instead.

Since $\large F$ is a derivative of $\large V$, we can think of $\large V$ as an antiderivative of $\large F$. We will want to find the potential function for conservative vector fields, and the following video shows how this is done via antiderivatives: