37336 Vector Calculus and Partial Differential Equations
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Subject handbook information prior to 2025 is available in the Archives.
Credit points: 6 cp
Result type: Grade and marks
Requisite(s): 35102 Introduction to Analysis and Multivariable Calculus OR 33230 Mathematics 2 OR 33290 Statistics and Mathematics for Science OR 33401 Introductory Mathematical Methods OR 37132 Introduction to Mathematical Analysis and Modelling
These requisites may not apply to students in certain courses. See access conditions.
Anti-requisite(s): 35335 Mathematical Methods
Description
This subject gives an introduction to the theories of vector calculus and partial differential equations, with an emphasis on applications from Electromagnetic Theory and Quantum Mechanics. Topics covered include vector field theory, integration of vector fields over lines and surfaces, vector calculus identities, the Laplacian operator, the operator calculus, Fourier series, Sturm-Liouville theory, Green's functions, the finite element method.
Subject learning objectives (SLOs)
Upon successful completion of this subject students should be able to:
| 1. | Demonstrate understanding of the theory of vector calculus and partial differential equations |
|---|---|
| 2. | Find analytic and numerical solutions to problems in vector calculus and PDEs |
| 3. | Apply mathematical tools to problems arising in electromagnetism and quantum mechanics |
| 4. | Effectively communicate the steps and reasoning used to arrive at the solution |
Course intended learning outcomes (CILOs)
This subject also contributes specifically to the development of following course intended learning outcomes:
- Demonstrate theoretical and technical knowledge of mathematical sciences including calculus, discrete mathematics, linear algebra, probability, statistics and quantitative management. (1.1)
- Evaluate mathematical and statistical approaches to problem solving, analysis, application, and critical thinking to make mathematical arguments, and conduct experiments based on analytical, numerical, statistical, algorithms to solve new problems. (2.1)
- Work autonomously or in teams to demonstrate professional and responsible analysis of real-life problems that require application of mathematics and statistics. (3.1)
- Design creative solutions to contemporary mathematical sciences-related issues by incorporating innovative methods, reflective practices and self-directed learning. (4.1)
- Use succinct and accurate presentation of reasoning and conclusions to communicate mathematical solutions, and their implications, to a variety of audiences, using a variety of approaches. (5.1)
Contribution to the development of graduate attributes
This Subject will contribute to the following Graduate Attributes:
1. Disciplinary Knowledge (1.2, 1.3)
2. Research, Inquiry and Critical Thinking (2.3)
3. Professional, Ethical and Social Responsiblity (3.1, 3.2)
4. Reflection, Innovation and Creativity (4.1)
5. Communication Skills (5.1)
Teaching and learning strategies
Subject content will be presented in a combination of online materials and in the workshops.
Workshops will take the form of an interactive session (2 hours per week) where the material is covered in depth.
Students are expected to revise the online material before each workshop.
Computer labs / Tutorials (2 hours per week) will focus on the practical implementation of mathematical skills in smaller groups.
Content (topics)
Part 1: Scalar and Vector Fields. Gradient, Divergence and Curl in Cartesian and in polar coordinates. Line and surface integrals, and their application to the calculation of physical quantities. The fundamental theorem and conservative fields. Flux of a vector field through a surface, circulation around a loop. The theorems of Green, Gauss and Stokes. Applications from electromagnetic theory, including Maxwell's equations.
Part 2: Separation of variables in different coordinate systems. Bessel functions, Legendre polynomials and spherical harmonics. Operators and eigenfunctions. Orthogonality and properties of eigenfunction solutions. Generalised Fourier methods for solving PDEs. The wave equation, the heat equation, Schrodinger’s equation. The Dirac Delta function, Green’s functions and solving inhomogeneous PDEs. Introduction to the Finite Element Method. Applications from Quantum theory.
Assessment
Assessment task 1: Skills Development Exercises
| Intent: | This assessment task will contribute to the following Graduate Attributes: 1. Disciplinary Knowledge (1.1, 1.2) 4. Reflection, Innovation, Creativity (4.1) 5. Communication Skills (5.1) |
|---|---|
| Objective(s): | This assessment task addresses subject learning objective(s): 1 and 2 This assessment task contributes to the development of course intended learning outcome(s): 1.1, 4.1 and 5.1 |
| Type: | Exercises |
| Groupwork: | Individual |
| Weight: | 20% |
| Criteria: | correct choice and use of problem solving strategies and procedures, accurate mathematical reasoning |
Assessment task 2: Mid-session test
| Intent: | This assesment task will contribute to the following Graduate Attributes: 1. Disciplinary Knowledge (1.2) 2. Research, Inquiry and Critical Thinking (2.3) 3. Professional, Ethical and Social Responsiblity (3.2) 5. Communication Skills (5.1) |
|---|---|
| Objective(s): | This assessment task addresses subject learning objective(s): 1, 2, 3 and 4 This assessment task contributes to the development of course intended learning outcome(s): 1.1, 2.1, 3.1 and 5.1 |
| Type: | Quiz/test |
| Groupwork: | Individual |
| Weight: | 40% |
| Criteria: | Correct choice and use of problem solving strategies and procedures, accurate mathematical reasoning. Ability to apply mathematical tools to unseen problems. |
Assessment task 3: Final Examination
| Intent: | The Assessment Task will contribute to the following Graduate Attributes: 1. Disciplinary Knowledge (1.3) 2. Research, Inquiry and Critical Thinking (2.3) 3. Professional, Ethical and Social Responsiblity (3.2) 5. Communication Skills (5.1) |
|---|---|
| Objective(s): | This assessment task addresses subject learning objective(s): 1, 2, 3 and 4 This assessment task contributes to the development of course intended learning outcome(s): 1.1, 2.1, 3.1 and 5.1 |
| Type: | Examination |
| Groupwork: | Individual |
| Weight: | 40% |
| Criteria: | correct choice and use of problem solving strategies and procedures, accurate mathematical reasoning. Ability to apply mathematical tools to real-world problems. |
Minimum requirements
Students must obtain at least 50% overall to pass the subject.
Recommended texts
Advice on textbooks will be given in the initial lecture. The following texts are recommended for additional reference:
- Kreyzig, E. Advanced Engineering Mathematics, 9th edition, John Wiley and Sons, 2006.
- Salas, S. L, Hille, E. & Etgen, G. J. Calculus: One and Several Variables, 10th edition, John Wiley & Sons, 2007. (ISBN 9780470132203)
- J Stewart, Calculus: Concepts and Contexts, 3rd ed., ISBN 0534 409830. Thomson Brooks/Cole Publishing Company, 2006 (Metric ed.) or ISBN 0534 409865 (non-metric edition published in 2005).