Advanced Methods: Transforms (MAST90067)
Graduate courseworkPoints: 12.5Not available in 2019
About this subject
Overview
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This subject develops the mathematical methods of applied mathematics and mathematical physics with an emphasis on integral transform and related techniques. An introduction is given to the calculus of variations and the Euler-Lagrange equation. Advanced complex contour integration techniques are used to evaluate and invert Fourier and Laplace transforms. The general theory includes convolutions, Green’s functions and generalized functions. The methods of Laplace, stationary phase, steepest descents and Watson’s lemma are used to asymptotically approximate integrals. Throughout, the theory is set in the context of examples from applied mathematics and mathematical physics such as the brachistochrone problem, Fraunhofer diffraction, Dirac delta function, heat equation and diffusion.
Intended learning outcomes
After completing this subject students should:
- have learned how the calculus of variations, transform methods and associated asymptotic analysis apply in a variety of areas in applied mathematics and mathematical physics;
- appreciate the role of advanced contour integration techniques of complex analysis and to be able to use these techniques to calculate transform integrals;
- understand the basic concepts of asymptotic evaluation of integrals, know how to implement Laplace’s method, stationary phase and steepest descents and appreciate their applicability and limitations;
- be familiar with the basic properties of generalized functions and Green’s functions in applied mathematics and mathematical physics and their applications;
- have the ability to pursue further studies in these and related areas.
Generic skills
In addition to learning specific skills that will assist students in their future careers in science, they will have the opportunity to develop generic skills that will assist them in any future career path. These include:
- problem-solving skills: the ability to engage with unfamiliar problems and identify relevant solution strategies;
- analytical skills: the ability to construct and express logical arguments and to work in abstract or general terms to increase the clarity and efficiency of analysis;
- collaborative skills: the ability to work in a team;
- time-management skills: the ability to meet regular deadlines while balancing competing commitments.
Last updated: 3 November 2022
Eligibility and requirements
Prerequisites
| Code | Name | Teaching period | Credit Points |
|---|---|---|---|
| MAST30021 | Complex Analysis |
Semester 1 (On Campus - Parkville)
Semester 2 (On Campus - Parkville)
|
12.5 |
Plus one of:
| Code | Name | Teaching period | Credit Points |
|---|---|---|---|
| MAST20030 | Differential Equations | Semester 2 (On Campus - Parkville) |
12.5 |
MAST30029 Partial Differential Equations (pre-2014)
Corequisites
None
Non-allowed subjects
No disallowed subject combinations among new-generation subjects.
Recommended background knowledge
It is recommended that students have completed at least one of the following:
| Code | Name | Teaching period | Credit Points |
|---|---|---|---|
| MAST30030 | Applied Mathematical Modelling | Semester 1 (On Campus - Parkville) |
12.5 |
| MAST30031 | Methods of Mathematical Physics | Semester 2 (On Campus - Parkville) |
12.5 |
Inherent requirements (core participation requirements)
The University of Melbourne is committed to providing students with reasonable adjustments to assessment and participation under the Disability Standards for Education (2005), and the Assessment and Results Policy (MPF1326). Students are expected to meet the core participation requirements for their course. These can be viewed under Entry and Participation Requirements for the course outlines in the Handbook.
Further details on how to seek academic adjustments can be found on the Student Equity and Disability Support website: http://services.unimelb.edu.au/student-equity/home
Last updated: 3 November 2022
Assessment
Additional details
Up to 50 pages of written assignments (40%: two assignments worth 20% each, due mid and late in semester), a 3 hour written examination (60%, in the examination period).
Last updated: 3 November 2022
Dates & times
Not available in 2019
Time commitment details
170 hours
Last updated: 3 November 2022
Further information
- Texts
Prescribed texts
None
Recommended texts and other resources
Carl M. Bender and Steven A. Orszag, Advanced mathematical methods for scientists and engineers: Asymptotic methods and perturbation theory. Springer. (1999).
George F. Carrier, Max Krook, and Carl E. Pearson, Functions of a Complex Variable: Theory and Technique, SIAM (2005). - Related Handbook entries
This subject contributes to the following:
Type Name Course Master of Science (Mathematics and Statistics) Course Ph.D.- Engineering Course Master of Philosophy - Engineering Course Doctor of Philosophy - Engineering Informal specialisation Mathematics and Statistics - Available through the Community Access Program
About the Community Access Program (CAP)
This subject is available through the Community Access Program (also called Single Subject Studies) which allows you to enrol in single subjects offered by the University of Melbourne, without the commitment required to complete a whole degree.
Please note Single Subject Studies via Community Access Program is not available to student visa holders or applicants
Entry requirements including prerequisites may apply. Please refer to the CAP applications page for further information.
- Available to Study Abroad and/or Study Exchange Students
Last updated: 3 November 2022