64613 ALGEBRA AND CALCULUS II

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

Contents

STAFFING:

Examiner: D. MANDER
Moderator: P. CRETCHLEY

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:

  • d a Taylor polynomial approximation for a function of one variable and
    an expression for the error;
  • w a direction field for a simple first order differential equation;
  • solve separable differential equations;
  • solve first order linear differential equations;
  • find series solutions of first order differential equations;
  • use Euler's method to find numerical solutions of first order
    differential equations;
  • solve second order linear homogeneous and inhomogeneous
    equations with constant coefficients;
  • apply knowledge of differential equations to solve real world
    problems;
  • sketch the curve described by a vector function of one
    variable in three dimensions;
  • find the position vector function description of a given
    curve;
  • differentiate a vector function of one variable and describe
    its geometrical significance;
  • 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;
  • sketch simple surfaces from the above;
  • find partial derivatives of vector and scalar functions of
    many variables, and interpret them geometrically;
  • use the chain rule for partial derivatives;
  • find and identify local maxima, minima and critical points in
    functions of two variables;
  • find the directional derivative of a scalar field at a given
    point in a given direction;
  • find the gradient of a scalar field;
  • find the divergence and curl of a vector field;
  • determine whether a vector field is conservative and if so
    find the scalar potential for the field;
  • calculate line and work integrals;
  • determine if a line integral is independent of path;
  • evaluate double integrals and use double integrals to
    determine areas, volumes;
  • reverse the order of integration of double integrals;
  • change variables in double integrals;
  • apply Green's theorem;
  • apply techniques of multiple integration to calculations of
    mass, area, volume, centre of mass, moments of inertia;
  • demonstrate an understanding of the concepts of vector space,
    spanning set, linear independence, basis, dimension, column
    space, rank, null space, nullity;
  • find a basis for a vector space given a spanning set; and use
    this to determine if a vector is contained in the space;
  • 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;
  • demonstrate an understanding of the concepts of inner product,
    norm, angle and orthogonality;
  • find the projection of a vector onto a vector space and the
    residual;
  • apply the concept of projections to least squares fitting of
    curves to data points;
  • demonstrate an understanding of the concepts of linear
    transformations, standard matrix, linear operations and
    invariant lines;
  • apply the concepts above to Markov chain problems;
  • find the eigenvalues bases for the eigenspaces, of 2 x 2 and 3
    x 3 matrices;
  • find where possible a diagonalising matrix for given 2 x 2 or
    3 x 3 matrices.
  • solve systems of first order linear differential equations
    using diagonalisation;
  • find powers of diagonalisable matrices;
  • apply the properties of the eigenvalues and eigenvectors of
    symmetric matrices to quadratic forms;

    TOPICS:

     Description                                                    Weighting(%)
      

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

      

    2. 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 integraton
-    Areas and volumes
-    Change of variables in double integrals
-    Green's theorem
-    Triple integrals, order of integration
-    Volumes, moments of intertia, and centres of mass of
  laminas and solids.

      

    3. 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:

    Hughes-Hallett, D. & Gleason, A. & McCallum, W. 1998, Calculus.
    Single and Multivariable    , 2nd edn., Wiley, New York.
    NOTE: Hughes-Hallett may be purchased in separate single and
    multivariouable 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
Lectures                                      	56
Tutorials/Workshops                           	28
Private Study                                 	72
Examinations                                  	3
Assessments                                   	16

    ASSESSMENT DETAILS:

    No  *F/S Marks     Due        Description                              Wtg(%)    LBL WWW
1   S    5.00      03/09/99  WEEKLY HOMEWORK                           5.00      Y   N
2   S    50.00     03/09/99  ASSIGNMENT 1                              10.00     Y   N
3   S    50.00     22/10/99  ASSIGNMENT 2                              10.00     Y   N
4   S              END S2    3 HOUR RESTRICTED EXAMINATION             75.00     N   N

    *F=Formative, S=Summative

    OTHER REQUIREMENTS:

    1    Students  must  complete 80% of tutorial  exercises  and/or  mid-
     semester assessments to the satisfaction of the Examiner to  pass
     the unit.
2    To   obtain   a   pass  in  the  unit,  students   must   perform
     satisfactorily in all aspects of assessment.
3    The  due date for assessments is the day on which they are to  be
     despatched  to  the  Faculty  of Sciences.  Assignments  must  be
     received by 5pm on the due date.
4    Students  MUST  retain a copy of all assignments  which  must  be
     produced if and when required by the Examiner.
5    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.
6    Restricted   Examination:   a  restricted   examination   is   an
     examination   where  only  those  materials  specified   in   the
     examination paper are permitted during the examination.
    This information is accurate as at 17/11/99