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.

单词	VisualizationsOfExteriorForms
释义	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.
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