Standard syllabus
Advanced engineering mathematics · Graduate · Math
Learning objectives from the Advanced engineering mathematics 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 Advanced Engineering Mathematics — scope drawn from open advanced engineering mathematics outlines and typical US graduate engineering math syllabi.
Integral transforms and complex methods
- Laplace and Fourier transforms: advanced properties and inversion — Laplace and Fourier transforms: advanced properties and inversion
- Mellin transforms (introduction) — Mellin transforms (introduction)
- Complex contour integration for transform inversion — Complex contour integration for transform inversion
- Wiener–Hopf methods (introduction) — Wiener–Hopf methods (introduction)
- Discrete transforms — Discrete transforms
- FFT applications at scale — FFT applications at scale
Special functions and orthogonal expansions
- Bessel, Legendre, and Hermite functions — Bessel, Legendre, and Hermite functions; recurrence and orthogonality
- Sturm–Liouville problems — Sturm–Liouville problems
- Eigenfunction expansions — Eigenfunction expansions
- Green's functions for boundary-value problems — Green's functions for boundary-value problems
- Multidimensional separation of variables — Multidimensional separation of variables
- Asymptotics of special functions (introduction) — Asymptotics of special functions (introduction)
PDE methods for engineers
- Classification — Classification
- Canonical forms of linear PDEs — Canonical forms of linear PDEs
- Method of characteristics for first-order systems — Method of characteristics for first-order systems
- Variational formulations of elliptic problems (introduction) — Variational formulations of elliptic problems (introduction)
- Integral equation formulations (introduction) — Integral equation formulations (introduction)
- Well-posedness — Well-posedness
- Engineering interpretation of boundary conditions — Engineering interpretation of boundary conditions
Computational engineering mathematics
- Spectral methods for PDEs (introduction) — Spectral methods for PDEs (introduction)
- Finite element — Finite element
- Finite volume overview for multiphysics — Finite volume overview for multiphysics
- High-performance computing for large simulations — High-performance computing for large simulations
- Uncertainty propagation in engineering models — Uncertainty propagation in engineering models
- Coupled field problems: thermal–structural, electro-mechanical (overview) — Coupled field problems: thermal–structural, electro-mechanical (overview)
Case studies
- Signal and image processing pipelines using transform methods — Signal and image processing pipelines using transform methods
- Structural dynamics — Structural dynamics
- Modal superposition — Modal superposition
- Electromagnetics — Electromagnetics
- Waveguides (mathematical models) — Waveguides (mathematical models)
- Transport phenomena — Transport phenomena
- Diffusion with sources — Diffusion with sources
- Benchmark problems linking analytical — Benchmark problems linking analytical
- Numerical solutions — Numerical solutions
Learning objectives
Click an objective for study materials.
Integral transforms and complex methods
- Laplace and Fourier transforms: advanced properties and inversion — Laplace and Fourier transforms: advanced properties and inversion
- Mellin transforms (introduction) — Mellin transforms (introduction)
- Complex contour integration for transform inversion — Complex contour integration for transform inversion
- Wiener–Hopf methods (introduction) — Wiener–Hopf methods (introduction)
- Discrete transforms — Discrete transforms
- FFT applications at scale — FFT applications at scale
Special functions and orthogonal expansions
- Bessel, Legendre, and Hermite functions — Bessel, Legendre, and Hermite functions; recurrence and orthogonality
- Sturm–Liouville problems — Sturm–Liouville problems
- Eigenfunction expansions — Eigenfunction expansions
- Green's functions for boundary-value problems — Green's functions for boundary-value problems
- Multidimensional separation of variables — Multidimensional separation of variables
- Asymptotics of special functions (introduction) — Asymptotics of special functions (introduction)
PDE methods for engineers
- Classification — Classification
- Canonical forms of linear PDEs — Canonical forms of linear PDEs
- Method of characteristics for first-order systems — Method of characteristics for first-order systems
- Variational formulations of elliptic problems (introduction) — Variational formulations of elliptic problems (introduction)
- Integral equation formulations (introduction) — Integral equation formulations (introduction)
- Well-posedness — Well-posedness
- Engineering interpretation of boundary conditions — Engineering interpretation of boundary conditions
Computational engineering mathematics
- Spectral methods for PDEs (introduction) — Spectral methods for PDEs (introduction)
- Finite element — Finite element
- Finite volume overview for multiphysics — Finite volume overview for multiphysics
- High-performance computing for large simulations — High-performance computing for large simulations
- Uncertainty propagation in engineering models — Uncertainty propagation in engineering models
- Coupled field problems: thermal–structural, electro-mechanical (overv... — Coupled field problems: thermal–structural, electro-mechanical (overview)
Case studies
- Signal and image processing pipelines using transform methods — Signal and image processing pipelines using transform methods
- Structural dynamics — Structural dynamics
- Modal superposition — Modal superposition
- Electromagnetics — Electromagnetics
- Waveguides (mathematical models) — Waveguides (mathematical models)
- Transport phenomena — Transport phenomena
- Diffusion with sources — Diffusion with sources
- Benchmark problems linking analytical — Benchmark problems linking analytical
Definitions and structure
- Laplace and Fourier transforms: advanced properties and inversion — Laplace and Fourier transforms: advanced properties and inversion
- Mellin transforms (introduction) — Mellin transforms (introduction)
- Complex contour integration for transform inversion — Complex contour integration for transform inversion
- Wiener–Hopf methods (introduction) — Wiener–Hopf methods (introduction)
- Discrete transforms — Discrete transforms
Proofs and reasoning
- FFT applications at scale — FFT applications at scale
- Bessel, Legendre, and Hermite functions — Bessel, Legendre, and Hermite functions; recurrence and orthogonality
- Sturm–Liouville problems — Sturm–Liouville problems
- Eigenfunction expansions — Eigenfunction expansions
- Green's functions for boundary-value problems — Green's functions for boundary-value problems
Abstraction and generalization
- Multidimensional separation of variables — Multidimensional separation of variables
- Asymptotics of special functions (introduction) — Asymptotics of special functions (introduction)
- Classification — Classification
- Canonical forms of linear PDEs — Canonical forms of linear PDEs
- Method of characteristics for first-order systems — Method of characteristics for first-order systems
Modeling and computation
- Apply laplace and fourier transforms: advanced properties and inversi... — Apply laplace and fourier transforms: advanced properties and inversion in engineering contexts
- Apply mellin transforms (introduction) in engineering contexts — Apply mellin transforms (introduction) in engineering contexts
- Apply complex contour integration for transform inversion in engineer... — Apply complex contour integration for transform inversion in engineering contexts
- Apply wiener–hopf methods (introduction) in engineering contexts — Apply wiener–hopf methods (introduction) in engineering contexts
- Apply discrete transforms in engineering contexts — Apply discrete transforms in engineering contexts
Data and technology
- Use software to explore advanced engineering mathematics problems num... — Use software to explore advanced engineering mathematics 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
- FFT applications at scale — FFT applications at scale
- Bessel, Legendre, and Hermite functions — Bessel, Legendre, and Hermite functions; recurrence and orthogonality
- Sturm–Liouville problems — Sturm–Liouville problems
- Eigenfunction expansions — Eigenfunction expansions
- Green's functions for boundary-value problems — Green's functions for boundary-value problems
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.
- Integral transforms and complex methods
Laplace and Fourier transforms: advanced properties and inversion
Coming soon - Special functions and orthogonal expansions
Bessel, Legendre, and Hermite functions; recurrence and orthogonality
Coming soon - PDE methods for engineers
Classification
Coming soon - Computational engineering mathematics
Spectral methods for PDEs (introduction)
Coming soon - Case studies
Signal and image processing pipelines using transform methods
Coming soon
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$1,162 · Advanced engineering mathematics · 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.