Smooth manifolds

Compatible smooth coordinate charts support differentiation on a manifold.

Different ways to understand it

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Coordinate geometry
Coordinates translate geometric conditions into equations and algebraic calculations.
Differential forms
Alternating multilinear objects encode integration over oriented geometric pieces.
Gauss–Bonnet theorem
For a closed oriented surface, total Gaussian curvature relates to its Euler characteristic.
Tangent spaces
Tangent vectors at a point form a linear space describing infinitesimal directions.
Arc length
Integrating speed measures a sufficiently regular curve's length.