Week 3 Synthesis
Central Question
Can I explain how local coordinates give way to intrinsic calculus, curvature and boundary laws?
Why This Matters
The week should reorganize familiar calculus into a coherent geometric language while keeping topology, smoothness and metric structure distinct.
Learning Prompt
Consolidate multivariable calculus → local Euclidean structure → charts and atlases → manifolds → tangent and cotangent spaces → differentials and pullbacks → differential forms → exterior derivative → curvature and geodesics → generalized Stokes. Ask me to explain the purpose of each construction and what extra structure it requires. Use a sphere, cylinder and punctured plane to expose gaps in my mental model. Connect forms with vector calculus and geometry with mechanics, but quiz me through conceptual predictions and small examples rather than routine coordinate calculations.
Ideas
Examples
Questions
Connections
- From Surfaces to Manifolds
- Tangent Spaces
- Differential Forms and Integration
- Curvature
- Generalized Stokes Theorem
- Geometry Meets Physics
- Week overview