64613 ALGEBRA AND CALCULUS II

Year	No.	Offer	Mode	Description			Cred. Pts
99	64613 	S2  	X 	ALGEBRA AND CALCULUS II   	1.00

Contents

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

STAFFING:

PRE-REQUISITE(S)

64612

RATIONALE:

This  unit  follows directly from Algebra and Calculus I in developing
the  concepts  and  techniques  of calculus  and  linear  algebra  for
application to problems in engineering and science, or as a basis  for
higher study in mathematics.

SYNOPSIS:

This  unit  is  divided  into  four  modules.  Module  1  covers   the
introductory  concepts and techniques of the solution of  differential
equations  and  introduces Taylor series in  one  variable.  Module  2
extends  the  concepts  of  calculus and vectors  into  the  study  of
functions of several variables, curves and surfaces in space,  partial
derivatives,    maxima/minima   problems,   directional   derivatives,
gradient, vector fields, curl and divergence. Module 3 introduces  the
student  to  more abstract concepts of linear algebra covering  vector
spaces,  bases,  dimension, rank, the nature of  solutions  of  linear
algebraic equations, projections and transformations, eigenvalues  and
eigenvectors,  diagonalisation  and  systems  of  1st   order   linear
differential  equations. Module 4 examines integrals of  functions  of
several  variables including line integrals, double integrals,  triple
integrals, Green's theorem, moments and centres of mass.

OBJECTIVES:

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

1. find a Taylor polynomial approximation for a function of one
   variable and an expression for the error;
   
2. draw a direction field for a simple first order differential
   equation;
   
3. solve separable differential equations;
   
4. solve first order linear differential equations;
   
5. find series solutions of first order differential equations;
   
6. use Euler's method to find numerical solutions of first order
   differential equations;
   
7. solve second order linear homogeneous and inhomogeneous
   equations with constant coefficients;
   
8. apply knowledge of differential equations to solve real world
   problems;
   
9. sketch the curve described by a vector function of one
   variable in three dimensions;
   
10. find the position vector function description of a given
    curve;
    
11. differentiate a vector function of one variable and describe
    its geometrical significance;
    
12. demonstrate an understanding of the methods of describing a
    surface in space by position vector function of two variables,
    level surface and explicit formula, and to translate between
    the three;
    
13. sketch simple surfaces from the above;
    
14. find partial derivatives of vector and scalar functions of
    many variables, and interpret them geometrically;
    
15. use the chain rule for partial derivatives;
    
16. find and identify local maxima, minima and critical points in
    functions of two variables;
    
17. find the directional derivative of a scalar field at a given
    point in a given direction;
    
18. find the gradient of a scalar field;
    
19. find the divergence and curl of a vector field;
    
20. determine whether a vector field is conservative and if so
    find the scalar potential for the field;
    
21. calculate line and work integrals;
    
22. determine if a line integral is independent of path;
    
23. evaluate double integrals and use double integrals to
    determine areas, volumes;
    
24. reverse the order of integration of double integrals;
    
25. change variables in double integrals;
    
26. apply Green's theorem;
    
27. apply techniques of multiple integration to calculations of
    mass, area, volume, centre of mass, and moments of inertia
    etc.
    
28. demonstrate an understanding of the concepts of vector space,
    spanning set, linear independence, basis, dimension, column
    space, rank, null space, nullity;
    
29. find a basis for a vector space given a spanning set; and use
    this to determine if a vector is contained in the space;
    
30. relate information about the rank of coefficient and augmented
    matrices of a system of linear equations to the nature of the
    solution of the equations;
    
31. demonstrate an understanding of the concepts of inner product,
    norm, angle and orthogonality;
    
32. find the projection of a vector onto a vector space and the
    residual;
    
33. apply the concept of projections to least squares fitting of
    curves to data points;
    
34. demonstrate an understanding of the concepts of linear
    transformations, standard matrix, linear operations and
    invariant lines;
    
35. apply the concepts above to Markov chain problems;
    
36. find the eigenvalues and bases for the eigenspace, of 2 x 2
    and 3 x 3 matrices;
    
37. find where possible a diagonalising matrix for given 2 x 2 or
    3 x 3 matrices.
    
38. solve systems of first order linear differential equations
    using diagonalisation;
    
39. find powers of diagonalisable matrices;
    
40. apply the properties of the eigenvalues and eigenvectors of
    symmetric matrices to quadratic forms;
    

TOPICS:

 Description                                                    Weighting(%)

Differential Equations                                                25.00
-    Direction fields
-    First order linear
-    Series solution
-    Taylor series
-    Euler's method
-    Second order linear with constant coefficients
-    Applications

Multivariable calculus                                                50.00
-    Curves in space, vector functions
-    Geometrical interpretation of derivatives of vector
     functions
-    Surfaces in space, functions of several variables
-    Partial differentiation
-    Geometrical interpretation of partial derivatives
-    Maxima/minima problems
-    Directional derivatives, gradient of scalar fields
-    Vector fields, conservative fields, curl and divergence
-    line and work integrals
-    independence of path
-    double integrals, order of integration
-    areas and volumes
-    change of variables in double integrals
-    Green's theorem
-    Triple integrals, order of integration
-    Volumes, moments of inertia, and centres of mass of
     laminas and Solids.

Linear Algebra                                                        25.00
-    Vector spaces, spanning sets, bases, linear
     independence, dimension
-    Column and row space, rank, null space, nullity
-    Linear algebraic equations
-    Inner products, norm, orthogonality
-    Projections, least squares fitting
-    Linear transformations and operators
-    Markov chains
-    Eigenvalues and eigenvectors, diagonalisation
-    Systems of first order differential equations
-    Powers of a matrix
-    Symmetric matrices, quadratic forms

TEXT and MATERIALS to be PURCHASED:

NOTE: Hughes-Hallett may be purchases in separate single and
multivariable volumes.

Larson, R. & Edwards, B. 1996, Elementary Linear Algebra, 3rd edn.,
Heath, Massacheusetts.

RECOMMENDED REFERENCE MATERIALS:

Anton, H. & Rorres, C. 1994, Elementary Linear Algebra - Applications
Version, 7th edn., John Wiley, New York.

Hughes-Hallett, D. & Gleason, A. & McCallum, W. 1998, Calculus.
Single and Multivariable, Student Solution Manual, 2nd edn., Wiley,
New York.

Kreysig, E. 1993, Advanced Engineering Mathematics, 7th edn., Wiley,
New York.

Larson, A., Hostetler, R. & Edwards, B. 1994, Calculus, 5th edn.,
D.C. Heath.

Larson, R. & Edwards, B. 1996, Student Solutions Guide - Elementary
Linear Algebra, ISBN: 0-660-30643-5.

Larson, R. & Edwards, B. 1996, Technology Keystroke Guide: Elementary
Linear Algebra, Heath, ISBN: 0-669-39645-1.

Matlab, 1997 version 5, Student Edition, CD and Users Guide,
Prentice Hall.

STUDENT WORKLOAD REQUIREMENTS:

	ACTIVITY				HOURS
Private Study                                 	145
Examinations                                  	3
Assessments                                   	16

ASSESSMENT DETAILS:

No  *F/S Marks     Due        Description                              Wtg(%)    LBL WWW
1   S    50.00     03/09/99  ASSIGNMENT 1                              12.50     Y   N
2   S    50.00     22/10/99  ASSIGNMENT 2                              12.50     Y   N
3   S              END S2    3 HR RESTRICTED EXAMINATION               75.00     N   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 day on which they are to  be
     despatched  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.
4    In  accordance  with  University's  Assignment  Extension  Policy
     (Regulation  5.9), the examiner of a unit may grant an  extension
     of  the  due  date of an assignment in extenuating circumstances.
     This  policy  may  be  found in the USQ  Handbook,  the  Distance
     Education  Study  Guide and the Faculty of Sciences'  Orientation
     Handbook for new on-campus students. All students are advised  to
     study and follow the guidelines associated with this policy.
5    Restricted   Examination:   a  restricted   examination   is   an
     examination   where  only  those  materials  specified   in   the
     examination paper are permitted during the examination.