Tag: Manifolds

• Differentiable Manifolds: Charts and Atlases That Bring Calculus to Curved SpacesWe introduce charts, transition maps and maximal atlases, build explicit atlases on the sphere, projective space and the torus, and define C^k functions and maps.MathematicsManifold TheoryGraduateManifoldsChartsAtlasesDifferentiable structureProjective space~30 min
• Tangent Spaces and the Tangent Bundle: Three Ways to Define Velocity on a Curved SpaceWe build the tangent space three ways — curves, coordinate components, derivations — prove them naturally isomorphic via Hadamard's lemma, and construct the tangent bundle.MathematicsManifold TheoryGraduateManifoldsTangent spaceTangent bundleDifferentialDerivations~36 min