Differential forms

Alternating multilinear objects encode integration over oriented geometric pieces.

Different ways to understand it

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Covariant derivatives
A connection differentiates vector fields while accounting for changing tangent spaces.
Smooth manifolds
Compatible smooth coordinate charts support differentiation on a manifold.
Arc length
Integrating speed measures a sufficiently regular curve's length.
Curvature and torsion
Curvature measures tangent turning; torsion describes departure from a plane in space.
Gaussian curvature
The product of principal curvatures measures intrinsic surface curvature.