## Syllabus

**Background material on topological spaces**
- Open and closed sets,
- Toplogies and topological spaces
- Continuous functions
~~Supspace and quotient topologies~~
~~Bases~~
~~Compact sets~~

**Basic concept of a manifold **
- Smooth functions and diffeomorphisms
- Coordinate charts, compatibility of charts
- Atlases, equivalence of atlases, maximal atlases
- Hausdorff condition
- Definition of a manifold
- Induced topology, open, closed and compact subsets

**Smooth maps and partitions of unity**
- Smooth functions, bump functions, existence
- Smooth maps between manifolds
- Submersions, Immersions, embeddings
- Partitions of unity
- Manifolds are embeddable in euclidean space

**Vectors and vector fields**
- Tangent vectors as curves, derivations, and via local coordinates
- Tangent space
- Derivative of a smooth map
~~Tangent bundle~~
~~Vector fields as smooth sections and derviations~~
~~Flow of a vector field~~
~~Lie brackets as commutators and via flows~~
~~Frobenius Theorem~~

~~Differential Forms~~
~~Covectors and the Cotangent bundle~~
~~Pullback of a form~~
~~Exterior algebra and wedge product~~
~~Exterior derivative~~
~~Closed and exact forms~~

~~Integration~~
~~Integration of a top degree form~~
~~Integration along submanifolds~~
~~Stokes theorem~~