Lectures related to Smooth maps and differentials: Smooth maps

Covers the topology of Riemann surfaces, focusing on orientation and orientability.

Explores tangent spaces as free movement directions on submanifolds, offering a geometrically satisfying linearization notion.

Explores fixed-rank matrices as an embedded submanifold, focusing on constructing local defining functions and efficiently computing tangent vectors.

Introduces smooth manifolds, emphasizing the importance of submanifolds of linear spaces.

Explores smooth manifolds through defining functions and submanifolds, highlighting the importance of smoothness.

Covers manifolds, charts, compatibility, and submanifolds with smooth analytic equations.

Introduces differential forms on manifolds, covering tangent bundles and intersection pairings.

Covers tangent spaces and submersions in differential geometry, emphasizing vector spaces and differentiable structures.