---
tag: 9T6ZEV4125
kind: concept
name: Single layer potential
msc: [31B10, 31A10]
specializes: [JHKSF3XFY2]
sources:
  - url: https://en.wikipedia.org/wiki/Newtonian_potential
---

## Summary

A single layer potential is the potential produced by a charge spread over a surface with a given density.
It is the integral of the [[fundamental solution of the Laplacian|BSJ8X2XRN9]] against the density times surface measure on the boundary of a domain.
It is [[harmonic|0DEB0JMN0W]] off the surface and continuous across it, while its normal derivative jumps by the density, as the [[jump relations|J58YA8QNNX]] say.
Together with the [[double layer potential|EG8Z3FWV9A]], it turns boundary value problems for the Laplace equation into integral equations on the boundary.

## Explanations

### L1 human

Paint electric charge onto a thin shell, with more charge in some places and less in others.
The electric potential that this painted charge creates in space is a single layer potential.
Away from the shell it is a smooth harmonic function, and when you cross the shell its value does not jump, but its slope does.
For a sphere in space with charge spread evenly, the potential is constant inside the sphere and falls off like $1/\abs{x}$ outside it.