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Stokes' Theorem provides interesting results for special forms

  • A form $\alpha$ is said to be closed if its exterior derivative is zero: $\mathrm{d}\alpha=0$
  • A form $\alpha$ is said to be exact if it is the exterior derivative of some other form: $\alpha=\mathrm{d}(\beta)$

And since $\mathrm{d}^2=0$, exact forms are immediately closed. Moreover:

  • A closed form always vanishes in submanifolds that are a boundary

  • An exact form always vanishes in closed submanifolds, no matter if they are boundary of something else

This is the very beginning of de Rham Cohomology