---
tag: 1Y4WX4XQWJ
kind: concept
name: Hodge star operator
msc: [58A10, 15A75, 53C20]
sources:
  - url: https://en.wikipedia.org/wiki/Hodge_star_operator
same_as:
  wikidata: Q1622692
  wikipedia: Hodge star operator
---

## Summary

The Hodge star operator is a linear map that sends $k$-forms to $(n-k)$-forms.
It is defined on an oriented $n$-dimensional inner product space and, pointwise, on an oriented [[Riemannian manifold|7CBT4SM83Y]].
It sends a form to its "orthogonal complement", using the metric and the orientation.
With it one defines the codifferential, the Hodge Laplacian and harmonic forms, and it underlies Hodge theory and Poincaré duality for de Rham cohomology.

## Explanations

### L1 human

In three-dimensional space, a small oriented plane piece and the arrow perpendicular to it carry the same information.
The Hodge star makes this exchange: it turns the plane spanned by $dx$ and $dy$ into the direction $dz$, and back.
In $n$ dimensions it exchanges $k$-dimensional pieces with $(n-k)$-dimensional ones.
For example, the cross product of two vectors in $\R^3$ is the Hodge star of their wedge product.