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- Introduction
- Vectors and oneforms
- Vectors
- Oneforms and dual vector space
- Tensors
- Metric tensor
- Basis vectors and oneforms
- Tensor algebra
- Change of basis
- Coordinate bases
- Isomorphism of vectors and oneforms
- Example Euclidean plane
- Differentiation and Integration
- Gradient of a scalar
- Gradient of a vector covariant derivative
- Christoel symbols
- Gradients of oneforms and tensors
- Evaluating the Christoel symbols
- Transformation to locally at coordinates
- Tensor Calculus, Part 2
- Introduction
- Orthonormal Bases, Tetrads, and Commutators
- Tetrads
- Commutators
- Connection for an orthonormal basis
- Number-Flux Vector and Stress-Energy Tensor
- Introduction
- Number-Flux Four-Vector for a Gas of Particles
- Lorentz Invariance of the Dirac Delta Function
- Stress-Energy Tensor for a Gas of Particles
- Uniform Gas of Non-Interacting Particles
- Parallel transport and geodesics
- Differentiation along a curve
- Parallel transport
- Geodesics
- Integrals of motion and Killing vectors
- Hamiltonian Dynamics of Particle Motion
- Introduction
- Geodesic Motion
- Separating Time and Space
- Hamiltonian mechanics and symplectic manifolds
- Extended phase space
- Reduction of order
