Handouts
Main
Interior, closure, and boundary: pdf.
Product topology: pdf.
Topologists' sine curve: pdf.
Hyperplane complements: pdf.
Gram-Schmidt and connectedness: pdf.
Compactness review: pdf.
Compactness criteria: pdf.
Completion of metric spaces: pdf.
Quotient vector spaces: pdf.
Real quadratic spaces: pdf.
Geometric approach to signature pdf.
Gluing topologies: pdf.
Construction of tensor products: pdf.
Calculations and further examples with tensor products: pdf.
Maps with tensor products: pdf.
Higher derivatives and multilinear maps: pdf.
Higher derivatives and multilinear Taylor's formula: pdf.
Bases of symmetric and exterior powers: pdf.
Tensor algebra and tensor pairings: pdf.
Topology of projective space: pdf.
Calculations with symmetric and exterior powers: pdf.
Paracompactness and local compactness: pdf.
Products of manifolds: pdf.
Atlases and differentiable structures: pdf.
Morse Lemma: pdf.
Manifolds with corners: pdf.
Quotients by group actions: pdf.
Orientations: pdf.
Global/local and bump functions: pdf.
Derivative mappings, parametric curves, and velocity vectors: pdf.
Tangent spaces and products: pdf.
Mobius strip in 3-space: pdf.
Local structure theorem: pdf.
Bijectivity and isomorphism: pdf.
Hodge star: pdf.
Submersions and transversality: pdf.
Construction of vector fields pdf.
Derivations and vector fields: pdf.
Modules and derivations: pdf.
Maps of bundles and O-modules: pdf.
Linear ODE: pdf.
Direct sum of bundles: pdf.
Equivalence of bundles and O-modules: pdf.
Integral curves: pdf.
Pullback and transition matrices: pdf.
Bundle operations: pdf.
Subbundles and quotient bundles: pdf.
Universal and normal bundles: pdf.
From integral curves to integral manifolds: pdf.
Example of integral manifold: pdf.
Determinant bundles: pdf.
Frobenius theorem and vector fields: pdf.
Operations with metrics: pdf.
Graph metric tensor: pdf.
Universe is not a 4-sphere: pdf.
Orientation on bundles and manifolds: pdf.
Coordinate-free d-map on forms: pdf.
The map-maker's paradox pdf.
Gauss map and scalar curvature: pdf.
How to compute integrals: pdf.
Stokes' theorem on Riemannian manifolds: pdf.
Maxwell's equations: pdf.
Stokes' theorem with corners: pdf.
Smooth hairy ball theorem pdf.
Covariant derivatives and parallel transport: pdf.