Tensor analysis
Graduate · Math
Syllabus focus
Standard syllabus · STEM / applied
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Topics typically covered
Standard syllabus
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
STEM / applied
Engineering and computational applications
- Constitutive modeling in solids and fluids
- Coordinate-free formulations in robotics and kinematics
- Numerical tensor calculus on unstructured grids (overview)
- Voigt notation and engineering tensor conventions
- Finite deformation and objective rates in nonlinear elasticity (introduction)
Software and case studies
- Symbolic tensor manipulation tools
- Voigt/Mandel conversions in FEM codes
- Anisotropic material orientation in composite models
- Visualization of tensor fields (stress, diffusion tensors)
- Benchmark problems in elasticity and heat conduction with anisotropy
Notes
Topics reflect common graduate tensor analysis syllabi at US universities, often taken by engineering and physics students alongside differential geometry courses.