Differential Geometry (8): Differential Forms — The Natural Language of Integration on Manifolds

Mon, 15 Nov 2021 09:00:00 +0000

In vector calculus on $\mathbb{R}^3$ , we have three derivative operations: gradient ( $\nabla f$ ), curl ( $\nabla \times F$ ), and divergence ( $\nabla \cdot F$ ). Each operates on a different type of object (scalar fields, vector fields). Two identities — $\nabla \times (\nabla f) = 0$ and $\nabla \cdot (\nabla \times F) = 0$ — sit there looking like happy coincidences. The three integral theorems (the fundamental theorem for line integrals, the classical Stokes’ theorem, the divergence theorem) appear unrelated and unmotivated except by their statements.

Differential-Forms on Chen Kai Blog

Differential Geometry (8): Differential Forms — The Natural Language of Integration on Manifolds