STEM / applied
Tensor analysis · Graduate · Math
Learning objectives from the Tensor analysis syllabus, grouped by unit. Click an objective for study materials.
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
Learning objectives
Click an objective for study materials.
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
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
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
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
Definitions and structure
- 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
Proofs and reasoning
- Raising/lowering indices — Raising/lowering indices
- Tensor products and contractions — Tensor products and contractions
- Symmetrization — Symmetrization
- Pseudotensors — Pseudotensors
- Orientation (introduction) — Orientation (introduction)
Abstraction and generalization
- 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
Modeling and computation
- Apply index notation and summation convention in engineering contexts — Apply index notation and summation convention in engineering contexts
- Apply change of basis in engineering contexts — Apply change of basis in engineering contexts
- Apply tensors as multilinear maps in engineering contexts — Apply tensors as multilinear maps in engineering contexts
- Apply contravariant components in engineering contexts — Apply contravariant components in engineering contexts
- Apply metric tensors in engineering contexts — Apply metric tensors in engineering contexts
Data and technology
- Use software to explore tensor analysis problems numerically — Use software to explore tensor analysis problems numerically
- Interpret computational results against analytic predictions — Interpret computational results against analytic predictions
- Build spreadsheets or scripts for routine calculations — Build spreadsheets or scripts for routine calculations
- Visualize functions, fields, or datasets tied to course topics — Visualize functions, fields, or datasets tied to course topics
- Connect course methods to lab, industry, or research workflows — Connect course methods to lab, industry, or research workflows
Problem-solving practice
- Raising/lowering indices — Raising/lowering indices
- Tensor products and contractions — Tensor products and contractions
- Symmetrization — Symmetrization
- Pseudotensors — Pseudotensors
- Orientation (introduction) — Orientation (introduction)
Multi-Unit Problems
Course-level sets that combine skills across study units (coming soon).
Browse Multi-Unit ProblemsWhat each unit includes
Open a unit below for full materials. Typical resources:
- Study guide
- Exam Strategy
- Common Mistakes
- Worksheets
- Word problems
- Mixed Practice
- Multi-Unit Problems
- Review
- Practice test
- Answer key
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
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$1,162 · Tensor analysis · 18 tutoring hrs
Study guides, worksheets, reviews, practice tests, and answer keys for 1 class. 18 tutoring hours (1 hr / week · semester). Bundle discount applied vs buying separately. Pay in full via Zelle.