Standard syllabus
Tensor analysis · Graduate · Math
Topics
Tensor algebra and calculus
- Index notation, summation convention, and change of basis
- Tensors as multilinear maps; covariant and contravariant components
- Metric tensors and raising/lowering indices
- Tensor products, contractions, and symmetrization
- Pseudotensors and orientation (introduction)
Calculus on manifolds
- Manifolds, charts, and tangent spaces (tensor viewpoint)
- Vector fields, differential forms, and exterior algebra
- Covariant derivative and connection coefficients
- Curvature tensor and geodesic equation
- Integration on manifolds and volume forms
Applications to physics
- Stress and strain tensors in continuum mechanics
- Electromagnetic field tensor and Maxwell equations
- General relativity: metric, Christoffel symbols, and Einstein equations (introduction)
- Invariants and principal directions in material models
- Transformation rules under orthogonal and Lorentz groups
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