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Differential Geometry

Curves, surfaces, curvature, manifolds: the language of general relativity.

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Advanced Curves, arc length and curvature Parametrised curves, the unit tangent, curvature as the turning rate. integrate sqrt(1 + 4x^2) dx from 0 to 1 Advanced Surfaces and Gaussian curvature Principal curvatures, Gauss's Theorema Egregium, geodesics. x^2 - y^2

Calculus on curved things. Curvature measures how a surface bends, geodesics are its straight lines, and manifolds let the same ideas run in any dimension. Einstein's equations are a statement in this language.

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Curves

Differential geometry of curves Curvature Arc length Frenet–Serret formulas Torsion of a curve Osculating circle Parametric equation

Surfaces

Differential geometry of surfaces Gaussian curvature Mean curvature Theorema Egregium First fundamental form Second fundamental form Geodesic Gauss–Bonnet theorem Minimal surface Developable surface

Manifolds

Manifold Differentiable manifold Tangent space Vector field Differential form Riemannian manifold Metric tensor Christoffel symbols Covariant derivative Riemann curvature tensor Parallel transport Lie group General relativity

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What is curvature?

For a curve, how fast its direction turns per unit length; for a surface, Gauss's combination of the two principal bendings, and Gauss's theorem says it can be measured from inside the surface without leaving it.

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