GR1 · Manifolds, tensors & geometry
A 20-module path through gr1 · manifolds, tensors & geometry, in study order from foundations to the frontier.
Modules
- Smooth Manifolds and Coordinate Charts
- Atlases, Transition Functions and Differentiable Structure
- Tangent Vectors and the Tangent Space
- Cotangent Space and One-Forms
- Vector Fields and Integral Curves
- General Tensors and the Tensor Product
- The Tensor Transformation Law
- Tensor Algebra: Contraction, Symmetrization and Antisymmetrization
- The Metric Tensor and the Line Element
- Metric Signature and Lorentzian Geometry
- Raising and Lowering Indices
- Tensor Densities and Weights
- The Invariant Volume Element
- Differential Forms and the Exterior Algebra
- The Wedge Product
- Exterior Derivative
- The Hodge Star Operator
- The Lie Derivative
- The Lie Bracket and Commutator of Vector Fields
- Pushforwards, Pullbacks and Diffeomorphisms