Tensor analysis
Graduate · Math
Syllabus focus
Topics typically covered
Click a topic for the full text and related unit practice.
Graduate Tensor Analysis — scope drawn from open tensor calculus notes and typical US graduate tensor analysis syllabi.
Tensor algebra and calculus
- Index notation and summation convention — Index notation and summation convention
- Change of basis — Change of basis
- Tensors as multilinear maps — Tensors as multilinear maps
- Contravariant components — Contravariant components
- Metric tensors — Metric tensors
- Raising/lowering indices — Raising/lowering indices
- Tensor products and contractions — Tensor products and contractions
- Symmetrization — Symmetrization
- Pseudotensors — Pseudotensors
- Orientation (introduction) — Orientation (introduction)
Calculus on manifolds
- Manifolds and charts — Manifolds and charts
- Tangent spaces (tensor viewpoint) — Tangent spaces (tensor viewpoint)
- Vector fields and differential forms — Vector fields and differential forms
- Exterior algebra — Exterior algebra
- Covariant derivative — Covariant derivative
- Connection coefficients — Connection coefficients
- Curvature tensor and geodesic equation — Curvature tensor and geodesic equation
- Integration on manifolds — Integration on manifolds
- Volume forms — Volume forms
Applications to physics
- Stress and strain tensors in continuum mechanics — Stress and strain tensors in continuum mechanics
- Electromagnetic field tensor — Electromagnetic field tensor
- Maxwell equations — Maxwell equations
- General relativity — General relativity
- Einstein equations (introduction) — Einstein equations (introduction)
- Invariants — Invariants
- Principal directions in material models — Principal directions in material models
- Transformation rules under orthogonal — Transformation rules under orthogonal
- Lorentz groups — Lorentz groups
Engineering and computational applications
- Constitutive modeling in solids and fluids — Constitutive modeling in solids and fluids
- Coordinate-free formulations in robotics — Coordinate-free formulations in robotics
- Kinematics — Kinematics
- Numerical tensor calculus on unstructured grids (overview) — Numerical tensor calculus on unstructured grids (overview)
- Voigt notation — Voigt notation
- Engineering tensor conventions — Engineering tensor conventions
- Finite deformation — Finite deformation
- Objective rates in nonlinear elasticity (introduction) — Objective rates in nonlinear elasticity (introduction)
Software and case studies
- Symbolic tensor manipulation tools — Symbolic tensor manipulation tools
- Voigt/Mandel conversions in FEM codes — Voigt/Mandel conversions in FEM codes
- Anisotropic material orientation in composite models — Anisotropic material orientation in composite models
- Visualization of tensor fields (stress, diffusion tensors) — Visualization of tensor fields (stress, diffusion tensors)
- Benchmark problems in elasticity — Benchmark problems in elasticity
- Heat conduction with anisotropy — Heat conduction with anisotropy
Study units
Each unit includes a study guide, worksheets, review, practice test, and answer key. One unit is free; subscribe for the full class.
- Tensor algebra and calculus
Index notation and summation convention
Coming soon - Calculus on manifolds
Manifolds and charts
Coming soon - Applications to physics
Stress and strain tensors in continuum mechanics
Coming soon - Engineering and computational applications
Constitutive modeling in solids and fluids
Coming soon - Software and case studies
Symbolic tensor manipulation tools
Coming soon
Notes
Graduate Tensor Analysis — scope drawn from open tensor calculus notes and typical US graduate tensor analysis syllabi. Topic outline: `content/topics/graduate/tensor_analysis.json`.