visualizations of exterior forms

There are (relatively) easy ways to visualize low-dimensional differential formsMathworldPlanetmath [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 dimensionsPlanetmathPlanetmath there’s only a one-dimensional space of 3-forms, and why a global one-form tells you about orientation.


  • 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 cohomologyMathworldPlanetmath). 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