Differential Topology Week 3: Cotangent bundles and differential forms

00:00:00 Covectors and tensors - dual spaces and tensor products
00:36:26 The cotangent bundle over a smooth manifold
00:38:48 Local triviality of fibre bundles
00:46:25 The vertical/horizontal and twisted structure of tangent/cotangent bundles
00:52:00 Differential 1-forms
00:58:45 The exterior derivative of a smooth function
01:01:25 Wedge products and the exterior algebra of differential k-forms
01:04:59 The determinant formula for evaluating differential k-forms
01:10:05 The exterior derivative of a differential k-form
01:15:45 Example: Converting integrals from cartesian to polar coordinates
01:20:11 Example: Exterior derivative of a differential 2-form