Advanced Methods: Transforms

Subject MAST90067 (2016)

Note: This is an archived Handbook entry from 2016.

Credit Points: 12.5
Level: 9 (Graduate/Postgraduate)
Dates & Locations:

This subject has the following teaching availabilities in 2016:

Semester 1, Parkville - Taught on campus.
Pre-teaching Period Start not applicable
Teaching Period 29-Feb-2016 to 29-May-2016
Assessment Period End 24-Jun-2016
Last date to Self-Enrol 11-Mar-2016
Census Date 31-Mar-2016
Last date to Withdraw without fail 06-May-2016


Timetable can be viewed here. For information about these dates, click here.
Time Commitment: Contact Hours: 36 hours comprising two 1-hour lectures and one 1-hour practice class per week.
Total Time Commitment:

170 hours

Prerequisites:
Subject
Study Period Commencement:
Credit Points:
Semester 1, Semester 2
12.50

Plus one of:

Subject
Study Period Commencement:
Credit Points:

MAST30029 Partial Differential Equations (pre-2014)

Corequisites:

None

Recommended Background Knowledge:

It is recommended that students have completed at least one of the following:

Subject
Study Period Commencement:
Credit Points:
Non Allowed Subjects:

No disallowed subject combinations among new-generation subjects.

Core Participation Requirements:

For the purposes of considering requests for Reasonable Adjustments under the Disability Standards for Education (Cwth 2005), and Students Experiencing Academic Disadvantage Policy, academic requirements for this subject are articulated in the Subject Description, Subject Objectives, Generic Skills and Assessment Requirements for this entry.

The University is dedicated to provide support to those with special requirements. Further details on the disability support scheme can be found at the Disability Liaison Unit website: http://www.services.unimelb.edu.au/disability/

Coordinator

Prof Paul Pearce

Contact

papearce@unimelb.edu.au

Subject Overview:

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.

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.
Assessment:

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).

Prescribed Texts:

None

Recommended Texts:

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).

Breadth Options:

This subject is not available as a breadth subject.

Fees Information: Subject EFTSL, Level, Discipline & Census Date
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.
Related Course(s): Doctor of Philosophy - Engineering
Master of Philosophy - Engineering
Master of Science (Mathematics and Statistics)
Related Majors/Minors/Specialisations: Mathematics and Statistics

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