Mathematics / advanced
Geometry, topology, and invariance
Understand local coordinates and global structure through tangent spaces, differential forms, curvature, and elementary topology.
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Inside this module
Module 01 advanced
Geometry, topology, and invariance
Understand local coordinates and global structure through tangent spaces, differential forms, curvature, and elementary topology.
- 01
Topological spaces and continuity
Describe nearness through open sets, and distinguish continuous maps from reversible changes of description.
25 min - 02
Compactness and connectedness
Separate finite-cover control from the impossibility of splitting a space into disjoint open regions.
26 min - 03
Tangent and cotangent spaces
Treat vectors as directional derivatives and covectors as linear measurements, with explicit coordinate-change rules.
28 min - 04
Differential forms and wedge products
Build oriented measurements from covectors, and make antisymmetry do the bookkeeping for integration.
27 min - 05
Exterior derivatives and Stokes' theorem
Relate interior variation to oriented boundary integrals, while distinguishing locally exact forms from global potentials.
29 min - 06
Riemannian metrics and geodesics
Add lengths to a smooth manifold and distinguish locally straight motion from a globally shortest route.
30 min - 07
Connections and curvature
Differentiate vector fields consistently, distinguish coordinate coefficients from curvature, and declare the curvature sign convention.
31 min - 08
Fundamental groups and coverings
Track loops up to continuous deformation, then use lifts to make winding and global obstructions calculable.
30 min - 09
Homology and Euler characteristic
Turn cycles and boundaries into computable invariants, with coefficients and finiteness assumptions stated explicitly.
31 min - 10
Fiber bundles and gauge geometry
Distinguish local product coordinates from global triviality, and interpret gauge choices as local frames rather than new physical states.
32 min