APP MTH 7106 - Optimal Functions & Nanomechanics III

North Terrace Campus - Semester 2 - 2014

Many problems in the sciences and engineering seek to find a shape or function that minimises or maximises some quantity. For example, an engineer may design a yacht's hull to minimise drag. And in nature, the shape that a complicated protein might adopt is determined in part by the lowest-energy state available to the protein during the folding process. The Calculus of Variations extends familiar calculus techniques to answer questions regarding optimal geometry or functions. The Calculus of Variations is applicable to almost all continuous physical systems, ranging through elasticity, solid and fluid mechanics, electro-magnetism, gravitation, quantum mechanics and string theory. In this course we will consider, in particular, problems from Nanoscience. Nanoscience is a multidisciplinary field at the nexus of physics, chemistry and engineering. Materials and systems that may be very well understood at the macroscale can often exhibit surprising phenomena at the nanoscale. Topic covered are: Classical Calculus of Variations problems such as the geodesic, catenary and brachistochrone; derivation and use of the Euler-Lagrange equations; multiple dependent variables (Hamiltons equations) and multiple independent variables (minimal surfaces); constrained problems, problems with variable end points and those with non-integral constraints; conservation laws and Noethers theorem; computational solutions using Eulers finite difference and Rayleigh-Ritz methods. Many of the examples considered will draw from continuum modelling of the intermolecular interaction potential utilizing special functions (such as gamma, beta, hypergeometric and generalized hypergeometric functions of two variables) and by application of Eulers elastica.

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Course Details

Course Code	APP MTH 7106
Course	Optimal Functions & Nanomechanics III
Coordinating Unit	Applied Mathematics
Term	Semester 2
Level	Postgraduate Coursework
Location/s	North Terrace Campus
Units	3
Contact	Up to 3 hours per week.
Prerequisites	MATHS 1012
Assumed Knowledge	MATHS 2102, MATHS 2101
Assessment	Ongoing assessment: 30%, Exam 70%

Course Staff

Course Coordinator: Dr Barry Cox

Course Timetable

The full timetable of all activities for this course can be accessed from .

Course Learning Outcomes

Apply the calculus of variations to find optimal solutions to problems.
Appreciate the derivation of many physical laws from variational principles.
Express interaction calculations using hypergeometric and other special functions.
Formulate models for nanoscale interactions.
Find optimal solutions to variational problems both analytically and numerically, as appropriate.
Explain of the role of applied mathematics in interdisciplinary research.

最新糖心Vlog Graduate Attribute	Course Learning Outcome(s)
Knowledge and understanding of the content and techniques of a chosen discipline at advanced levels that are internationally recognised.	1,2,3,4,5,6
The ability to locate, analyse, evaluate and synthesise information from a wide variety of sources in a planned and timely manner.	1,3,4,5
An ability to apply effective, creative and innovative solutions, both independently and cooperatively, to current and future problems.	4,5
Skills of a high order in interpersonal understanding, teamwork and communication.	2,6
A proficiency in the appropriate use of contemporary technologies.	3,5
A commitment to continuous learning and the capacity to maintain intellectual curiosity throughout life.	1,2,3,4,5,6
A commitment to the highest standards of professional endeavour and the ability to take a leadership role in the community.	6
An awareness of ethical, social and cultural issues within a global context and their importance in the exercise of professional skills and responsibilities.	6

Learning & Teaching Modes

The lecturer guides the students through the course material in 30 lectures. Students are expected to engage with the material in the lectures. Interaction with the lecturer and discussion of any difficulties that arise during the lecture is encouraged. Students are expected to attend all lectures, but lectures will be recorded to help with occasional absences and for revision purposes. In fortnightly tutorials students present their solutions to assigned exercises and discuss them with the lecturer and each other. Fortnightly homework assignments help students strengthen their understanding of the theory and their skills in applying it, and allow them to gauge their progress.

Workload

The information below is provided as a guide to assist students in engaging appropriately with the course requirements.

Workload
Activity	Quantity	Workload Hours
Lectures	30	90
Tutorials	6	18
Assignments	6	48
Total		156

Learning Activities Summary

Schedule
Week 1	Background	Introduction - Extrema - Vector Calculus refresher
Week 2	Fundamental	Euler-Lagrange equations - Autonomous systems
Week 3	Fundamental	More autonomous - Geodesics - Invariance
Week 4	Extension	Higher order derivatives - Special functions
Week 5	Extension / Nanomechanics	Several dependent variables - Van der Waals forces - Interaction potentials
Week 6	Nanomechanics	Hypergeometric functions - Nanotube oscillators
Week 7	Extension	Several independent variables - Numerical solutions - Ritz method
Week 8	Fundamental	Lagrange multipliers - Isoperimetric problems
Week 9	Fundamental	Multiple integral constraints - Natural boundary conditions - Curvature and the elastica
Week 10	Nanomechanics	Join regions for carbon nano最新糖心Vlog - Free endpoints - AFM cantilever
Week 11	Extension	Traversals - Broken extremals - Hamiltonian formulation
Week 12	Extension	Conservation laws - Classifying extrema

Tutorials at end of Weeks 2, 4, 6, 8, 10 and 12 will cover the material of the previous five lectures.

The 最新糖心Vlog's policy on Assessment for Coursework Programs is based on the following four principles:

Assessment must encourage and reinforce learning.
Assessment must enable robust and fair judgements about student performance.
Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
Assessment must maintain academic standards.

Assessment Summary

Assessment
Task	Type	Due	Weighting	Learning Outcomes
Examination	Summative	Examination Period	70 %	All
Assignments	Formative and Summative	Weeks 2, 4, 6, 8, 10 and 12	30 %	All

Assessment Related Requirements

An aggregate score of 50% is required to pass the course.

Assessment Detail

Assessment Detail
Task	Set	Due	Weighting
Assignment 1	Week 1	Week 2	5 %
Assignment 2	Week 3	Week 4	5 %
Assignment 3	Week 5	Week 6	5 %
Assignment 4	Week 7	Week 8	5 %
Assignment 5	Week 9	Week 10	5 %
Assignment 6	Week 11	Week 12	5 %

Submission

Homework assignments must be submitted on time with a signed assessment cover sheet. Late assignments will not be accepted. Assignments will be returned within two weeks. Students may be excused from an assignment for medical or compassionate reasons. Documentation is required and the lecturer must be notified as soon as possible.

Course Grading

Grades for your performance in this course will be awarded in accordance with the following scheme:

M10 (Coursework Mark Scheme)
Grade	Mark	Description
FNS		Fail No Submission
F	1-49	Fail
P	50-64	Pass
C	65-74	Credit
D	75-84	Distinction
HD	85-100	High Distinction
CN		Continuing
NFE		No Formal Examination
RP		Result Pending

Further details of the grades/results can be obtained from Examinations.

Grade Descriptors are available which provide a general guide to the standard of work that is expected at each grade level. More information at Assessment for Coursework Programs.

Final results for this course will be made available through .

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Maintained by:	Course Outlines Administrator

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