Regular surfaces

Regular surface charts have linearly independent coordinate tangent vectors.

Different ways to understand it

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Coordinate geometry
Coordinates translate geometric conditions into equations and algebraic calculations.
Curvature and torsion
Curvature measures tangent turning; torsion describes departure from a plane in space.
Regular curves
A regular curve has a nonzero velocity, allowing local tangent geometry.
Arc length
Integrating speed measures a sufficiently regular curve's length.
Covariant derivatives
A connection differentiates vector fields while accounting for changing tangent spaces.