---
tag: 360H7MD9RD
kind: concept
name: Current
msc: [49Q15, 58A25]
sources:
  - url: https://en.wikipedia.org/wiki/Current_(mathematics)
same_as:
  wikidata: Q3001145
  wikipedia: Current (mathematics)
---

## Summary

An $m$-dimensional current is a continuous linear functional on the smooth compactly supported [[differential $m$-forms|0E3G0GRPZ6]] of an open set of $\R^n$.
Currents generalise oriented surfaces, in the way that distributions generalise functions.
They carry a boundary operator and a notion of mass, and limits of surfaces of bounded area and boundary length exist among them.
This makes currents a main tool for the Plateau problem and other minimisation problems in geometric measure theory.

## Explanations

### L1 human

An oriented surface $S$ can be known by what it does to forms: it sends a form $\omega$ to the number $\int_S \omega$.
A current is any rule of this kind that is linear and continuous, even when no surface lies behind it.
For example, a point $p$ gives the $0$-current that sends a function $f$ to $f(p)$.
A curve traced twice gives the current "twice the curve", and a curve with the opposite orientation gives its negative.
Stokes' theorem becomes the definition of the boundary: the boundary of a current is the current that sends $\omega$ to the value on $d\omega$.