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Mathematics

Differential Forms and Integration on Manifolds

Quick fact

Differential forms are the objects that can be integrated on manifolds without a metric, and they make Stokes' theorem a single elegant formula: the integral of a form over the boundary of a manifold equals the integral of its exterior derivative over the whole manifold.

Why this is interesting

You know how to integrate functions over intervals and surfaces, but what does it mean to integrate over a space that is curved and has no global coordinate system? Differential forms are the key that makes integration on any manifold natural and coordinate-free.