64622 VECTOR ANALYSIS AND DIFFERENTIAL EQUATIONS

Year	No.	Offer	Mode	Description			Cred. Pts
98	64622 	S1  	D 	VECTOR ANAL & DIFF EQUATNS	1.00

Contents

• Staffing
• Pre-requisite(s)
• Rationale
• Synopsis
• Objectives
• Topics
• Texts
• Recommended References
• Student Workload
• Assessment Details
• Other Requirements

STAFFING:

PRE-REQUISITE(S)

64613/75613

RATIONALE:

This  unit  provides basic mathematical tools and modelling techniques
required for an understanding of the principles that permeate much  of
mathematical physics and engineering science.

SYNOPSIS:

This unit is broadly divided into two interrelated halves.

The  vector calculus half develops further the theory and applications
of  the  calculus  of  vectors from that  established  in  64613.  The
principles  of modelling material such as air or water are  introduced
in  one-dimensional dynamics. Then the application of vector  calculus
to  modelling the 3-D flow of a fluid material is developed along with
the  notions  of scalar and vector functions, directional derivatives,
divergence,  curl  and  Laplace's equation. The integral  theorems  of
Gauss  and  Stokes  highlight many facets  of  fluid  flow  and  other
physical   fields.   Many  practical  problems   involve   curvilinear
coordinates,  requiring expressions for vector differential  operators
in  general coordinate systems. An application is to flow in  a  thin,
twisting tube.

The  differential equations (DE's) half of the course  begins  with  a
formal  study of infinite series, leading to power series and Taylor's
theorem n-dimensions. Application is made to the classification
of  extrema  in  constrained optimisation problems. The  reduction  of
general  DE's  to  systems  of linear, first-order  DE's  is  used  to
visualise  dynamics  using the phase diagram and  to  introduce  fixed
points,  stability and limit cycles. Fourier analysis  introduces  the
use  of series expansion and integral transform methods which are then
employed in the solution of partial differential equations that appear
in models of many physical systems.

OBJECTIVES:

On completion of this unit, students will be able to:

1. determine coordinate vectors for curves and surfaces in space
   and their tangent lines and tangent planes;
   
2. evaluate line and surface integrals;
   
3. apply the integral theorems to calculate flux and work
   integrals;
   
4. determine scalar potentials for conservative fields.
   
5. find Fourier series for given functions;
   
6. solve partial differential equations by the technique of
   seperation of variables;
   
7. find Laplace transforms of given functions;
   
8. use Laplace transforms to solve constant coefficient
   differential equations and systems of differential equations;
   
9. apply Laplace and Fourier transform techniques to the solution
   of partial differential equations.
   

TOPICS:

 Description                                                    Weighting(%)

Modelling material in 1-D                                             8.00
Eulerian description of motion; conservation of mass;
car traffic; conservation of momentum; gas flow.

Modelling fluid flow needs vector calculus                            17.00
scalar & vector fields of fluid flow; material and
directional derivatives;
divergence in the conservation of mass;
vorticity is the curl of velocity;
conservation of momentum, pressure, Venturi effect;
Laplace's equation describes irrotational flow.

Vector integral theorems                                              17.00
circulation is a work integral;
scalar potentials lead to path independence
surface integrals measure flux;
Gauss' divergence theorem transforms volume integrals;
vorticity and circulation are related by Stokes' theorem.

Curvilinear coordinates help curved shapes                            8.00
unit vectors and scale factors vary in space, the gradient;
integral theorems determine divergence and curl;
coordinates for a long twisting tube;
flow in a pipe.

Infinite Series                                                       10.00
tests for convergence, absolute/conditional convergence;
power series, radius and interval of convergence;
Taylor series and truncation error;
dimensional Taylor's theorem and the classification of
extrema in constrained optimisation problems.

Systems of differential equations                                     12.00
the solution of inear DE's and the reduction of higher-order
linear DE's to first-order systems;
fixed points and phase potraits, specially in 2-D;
approximate solution of nonlinear, first-order DE's,
especially in the region of fixed points.

Fourier Analysis                                                      12.00
Fourier series for functions with arbitrary period;
half-range expansions;
Fourier integrals and transforms.

Partial Differential Equations                                        16.00
PDE's model physical systems; the heat equation;
the wave equation; Laplace's equation;
classification of PDE's.

TEXT and MATERIALS to be PURCHASED:

Roberts, A.J., A One-dimensional Introduction to Continuum
Mechanics, World Science.

RECOMMENDED REFERENCE MATERIALS:

Amazigo, J.C. & Rubenfeld, L.A. 1980, Advanced Calculus, John Wiley,
New York.

Greenberg, Michael P. 1978, Foundations of Applied Mathematics,
Prentice-Hall, NJ.

Kaplan, W. 1981, Advanced Mathematics for Engineers, Addison-Wesley,
Reading, Mass.

STUDENT WORKLOAD REQUIREMENTS:

	ACTIVITY				HOURS
Lectures                                      	56
Tutorials/Workshops                           	28
Private Study                                 	60
Examinations                                  	3
Assessments                                   	21

ASSESSMENT DETAILS:

No  *F/S Marks     Due        Description                              Wtg(%)    LBL WWW
1   S              20/03/98  ASSIGNMENT 1                              5.00      N
2   S              27/03/98  ASSIGNMENT 2                              5.00      N
3   S              08/05/98  ASSIGNMENT 3                              5.00      N
4   S              15/05/98  ASSIGNMENT 4                              5.00      N
5   S              END S1    3 HOUR RESTRICTED EXAMINATION             80.00     N

OTHER REQUIREMENTS:

1    To   obtain   a   pass  in  the  unit,  students   must   perform
     satisfactorily in all aspects of assessment.
2    The  due date for assessments is the date by which a student must
     despatch an assignment to the USQ. The onus is on the student  to
     provide proof of the despatch date, if requested by the Examiner.
3    Students  must  retain a copy of all assignments  which  must  be
     produced if and when required by the Examiner.
In accordance with University policy and Guidelines,
i    an  Examiner  may  grant  an extension of  the  due  date  of  an
     assignment in extenuating circumstances;
ii   no  assignments  will be accepted for assessment  purposes  after
     assignments  or  model  solutions have been  released  except  in
     extenuating circumstances;
iii  assignments  submitted after the due date without any extenuating
     circumstances  will  attract a penalty of  at  most  20%  of  the
     assignment mark for each working day late;
iv   students who submit an assignment after the due date and wish  to
     claim   extenuating   circumstances,  must  provide   documentary
     evidence with the assignment explaining the circumstances;
v      the  unit  examiner  shall consider  a  claim  for  extenuating
     circumstances and decide on the outcome;
vi   the  decision of the Dean shall be final in any dispute that  may
     arise in the implementation of these guidelines.
Restricted  Examination: a restricted examination  is  an  examination
where  only  those  materials specified in the examination  paper  are
permitted during the examination.