# visualizations of exterior forms

There are (relatively) easy ways to visualize low-dimensional differential forms^{} [1]:

A 1-form is locally like a stack of papers; given a vector, it returns a number: how many sheets the arrow pierces.

A 2-form takes a pair of arrows and returns the ”area” of the parallelogram they define.

A 3-form takes a triple of arrows and returns the ”volume” of the parallelliped they span. This explains why in three dimensions^{} there’s only a one-dimensional space of 3-forms, and why a global one-form tells you about orientation.

## References

- 1 Misner, Thorne, and Wheeler, “Gravitation”, Freeman, 1973.

Editorial note: Descriptions of these with pictures would be nice (especially for helping to visualize de Rham cohomology^{}). Maybe they would be better off in an attached entry, though.

Title | visualizations of exterior forms |
---|---|

Canonical name | VisualizationsOfExteriorForms |

Date of creation | 2013-03-22 15:28:12 |

Last modified on | 2013-03-22 15:28:12 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 6 |

Author | rmilson (146) |

Entry type | Definition |

Classification | msc 15A75 |

Classification | msc 58A10 |

Related topic | DifferentialForms |