scalar potential

Given a vector field , the scalar potential is defined as such that

Note

In some cases, mathematicians may use a positive sign in front of the gradient to define the potential ¹

only has a scalar potential when it is a gradient field and is conservative.

Finding Scalar Potential

We can find the scalar potential by computing the line integral. By the fundamental theorem of calculus for line integrals, if , then

We can choose a convenient curve from the point to . (And if we use the negative gradient definition of , we can just set at the end).

let

We can check that this vector field is indeed conservative ().

We can integrate it twice, one from to (), and another from to ().

On , and ,

On , and ,

Thus, , and

Footnotes

1. Session 63: Potential Functions | Multivariable Calculus | MIT OpenCourseWare ↩