Differential Topology Week 2: Manifolds and Tangent Bundles

An error was made in the video at 01:16:40: TS^2 is not homeomorphic to S^2 x R^2, as otherwise the hairy ball theorem would not be true. They are, however, locally homeomorphic, which gives the local trivializations of the tangent bundle TS^2. At 01:26:07, I said "tangent bundles" when I meant just general "bundles."

00:00:00 Recap of week 1
00:02:10 Topological invariance of dimension
00:03:29 Manifolds
00:17:25 Charts and atlases
00:36:30 Smooth structures
00:41:48 Directional derivatives
00:52:08 Tangent spaces, tangent vectors as directional derivatives
01:04:58 Tangent bundles
01:25:50 Discussion