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Week 03 — Geometry and Manifolds

Week 03 — Geometry and Manifolds

Central Question

How can calculus exist intrinsically on curved spaces?

The Story

Reorganize multivariable calculus around local descriptions, intrinsic derivatives, forms, and the geometry added by a metric.

\mathbb R^n\rightarrow\text{Surfaces}\rightarrow\text{Manifolds}\rightarrow\text{Tangent Spaces}\rightarrow\text{Differential Forms}\rightarrow\text{Curvature}\rightarrow\text{Stokes}

Sessions

Connections Beyond This Week

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Related

Session 01 — From Surfaces to Manifolds Session 02 — Charts and Atlases Session 03 — Tangent Spaces Session 04 — Differential Maps and Pushforwards Session 05 — Cotangent Spaces and Differential Forms Session 06 — Differential Forms and Integration Session 07 — Exterior Derivative Session 08 — Curvature Session 09 — Geodesics Session 10 — Generalized Stokes Theorem Session 11 — Geometry Meets Physics Session 12 — Week 3 Synthesis Week 02 — Topology Week 04 — Probability Information and Emergence Month 01 — From Familiar Mathematics to Modern Mathematics Exploration Map Mathematics Exploration Library

Referenced by

Month 01 — From Familiar Mathematics to Modern Mathematics Week 02 — Topology Session 01 — From Surfaces to Manifolds Session 02 — Charts and Atlases Session 03 — Tangent Spaces Session 04 — Differential Maps and Pushforwards Session 05 — Cotangent Spaces and Differential Forms Session 06 — Differential Forms and Integration Session 07 — Exterior Derivative Session 08 — Curvature Session 09 — Geodesics Session 10 — Generalized Stokes Theorem Session 11 — Geometry Meets Physics Session 12 — Week 3 Synthesis Week 04 — Probability Information and Emergence Geometry Mathematics Exploration Library