Interior, closure, and boundary:
pdf.
Topologists' sine curve:
pdf.
Hyperplane complements:
pdf.
Gram-Schmidt and connectedness:
pdf.
Compactness criteria:
pdf.
Completion of metric spaces:
pdf.
Quotient vector spaces:
pdf.
Real quadratic spaces:
pdf.
Geometric approach to signature
pdf.
Construction of tensor products:
pdf.
Calculations and further examples with tensor products:
pdf.
Maps with tensor products:
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.
Manifolds with corners:
pdf.
Quotients by group actions:
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.
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.
Direct sum of bundles:
pdf.
Equivalence of bundles and O-modules:
pdf.
Pullback and transition matrices:
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.