Applied Multivariate and Vector Calculus

1998/99 CALENDAR COPY:*Topics
covered include grad, div, curl and Laplacian in polar coordinates;
line and surface integrals; theorems of Gauss and Stokes; double and triple
integrals in various coordinate systems; extrema and Taylor series for
multivariate functions; differential geometry in Euclidean 3-space.*

[*Ed. note:* Repeated requests for input elicited no reply
from the course coordinator, so we have used the 1997/98 description here.]

This course involves the differentiation and integration of scalar and vector functions of (up to) 3 independent variables, with applications. Further topics include partial derivatives, curves and surfaces in cartesian, cylindrical, and spherical polar coordinate systems, single integrals, differential vector identities, line, surface, and volume integrals, and Green's theorem.

*Prerequisite: *One of AS/SC/MATH1010 3.0, AS/SC/

MATH1014 3.0,
AS/SC/AK/MATH1310 3.0, or:
AS/SC/

MATH1505 6.0 plus permission of the
Course Coordinator.

*Degree credit exclusions: *AS/SC/MATH2010 3.0,

AS/SC/AK/MATH2310 3.0.

**Coordinator: ** J. Darewych

**AS/SC/MATH2018 1.0 F**

**Applied Mathematics Module III**

*Note:* This course will be offered for the first time in 1999/2000.

**AS/SC/MATH2021 3.0 FLinear Algebra I
(Honours Version) **

1998/99 CALENDAR COPY:*Linear
equations, matrices, determinants, vector spaces and inner product
spaces. This course covers material similar to that in AS/SC/AK/MATH2221 3.0
but at a more advanced level. It is required in Honours Mathematics degrees.
*

Linear algebra is the study of vectors, matrices and linear transformations. These concepts are used in all areas of mathematics, in all branches of science and in the quantitative aspects of the social sciences. The content of this course is similar to that of MATH2221 3.0. This course, however, is more theoretical and covers additional topics in linear algebra and its applications. It therefore provides a solid background not only for courses in linear programming and statistics which use linear algebra but also for advanced theoretical mathematics courses such as abstract algebra and functional analysis.

The course begins with concrete topics such as the solution of linear systems of equations by Gauss-Jordan reduction, matrices and determinants. We then proceed to study the more abstract concepts of vector spaces, basis, dimension and inner products.

This is a rigorous mathematics course in which all definitions and proofs are presented in class. In addition, all concepts will be illustrated with examples and reinforced through computational homework problems. Moreover, students are expected to learn to construct proofs as the course progresses. Theoretical homework problems will be assigned, graded and counted towards the final grade. Most exam questions, however, will be computational in nature.

This is an honours course intended primarily for students intending to earn an honours degree in mathematics (other than Honours Mathematics for Commerce). However, all students who meet the prerequisites are welcome.

The text and grading scheme have not yet been determined.

Note that MATH1540 3.0 may not be taken for credit by anyone who is taking, or anyone who has taken, MATH2021.

*Prerequisite: *OAC algebra or any university mathematics course.

*Degree credit exclusions: *AS/SC/MATH1025 3.0,

AS/SC/MATH2000 6.0,
AS/SC/AK/MATH2221 3.0,

AK/MATH2220 6.0.

**Coordinator: ** Yuri Medvedev

**AS/SC/MATH2022 3.0 W
Linear Algebra II (Honours Version)**

1998/99 CALENDAR COPY:*Linear
transformations, eigenvalues, diagonalization, quadratic forms,
Markov chains and isometries. This course covers material similiar to that
in AS/SC/AK/ MATH2222 3.0 but
at a more advanced level. It is required in
Honours degrees in Mathematics and in Specialized Honours degrees in
Statistics.*

This course is a continuation of MATH2021 3.0. It covers more topics and applications than MATH2222 3.0 (see above, and add ``least squares approximations'' to the list). The high points of the course are the Cayley-Hamilton Theorem and the orthogonal diagonalization of symmetric matrices.

The text will be the same as for MATH2021 3.0.

The marking scheme has yet to be determined.

*Prerequisites: *AS/SC/MATH2021 3.0
or

permission of the course coordinator.

*Degree credit exclusions: *AS/SC/MATH2000 6.0,

AS/SC/AK/MATH2222 3.0, AK/MATH2220 6.0.

**Coordinator: ** T.B.A.

**AS/SC/AK/MATH2030 3.0 F
Elementary Probability**

(formerly part of MATH2030 6.0)
*Note that this course has moved to Fall term
from its customary Winter position.*

1998/99 CALENDAR
COPY:*Introduction
to the theory of probability as preparation for further study
in either mathematical or applied probability and statistics. Topics include
probability spaces, conditional probability, independence, random variables,
distribution functions, expectation, Chebyshev's inequality, common
distributions, moment-generating functions and limit theorems.*

This introductory course in probability is designed for those students who want more than a "cookbook" approach to the subject, for those who expect to take further courses in probability, mathematical statistics or stochastic processes, and for those majoring in Mathematics, Applied Mathematics or Statistics.

*Prerequisite: *One of AS/SC/MATH1010 3.0,

AS/SC/MATH1014 3.0, AS/SC/AK/MATH1310 3.0.

*Degree credit exclusion: *AS/SC/MATH2030 6.0.

**Coordinator: ** D. Salopek

**AS/SC/MATH2041 3.0 F
Symbolic Computation
Laboratory I**

1998/99 CALENDAR COPY:*An
introduction to symbolic computing in the Maple environment. Topics from
single-variable differential and integral calculus, including simple
ordinary differential equations, are covered. Both mathematical
understanding and applications are emphasized.
Enrolment is limited to about 30 per section.*

The course also provides students with an introduction to
numerical computation in mathematics using *Maple*. Classes take place
in a computer lab and students will spend most of their time working
independently, under the guidance of the instructor, on assigned projects.
The course mark will be based on the marks obtained on the
assigned projects. For a possible marking scheme and a list of projects
see `http://www.math.yorku.ca/Who/Faculty/Steprans/
Courses/2041.html`.

The projects in MATH2041 require a knowledge of single variable calculus, linear algebra and high school algebra. No programming knowledge is required.

*Prerequisites: *SC/AS/COSC1540 3.0, or equivalent

computing experience; one of AS/SC/MATH1010 3.0,

AS/SC/MATH1014 3.0, AS/SC/AK/MATH1310 3.0.

*Degree credit exclusion: *AS/SC/MATH2040 6.0.

**Coordinator: ** J. Steprans

[There should be a bar over the "a"
in Steprans,

but the editor's knowledge of HTML is not up to this.]

**AS/SC/MATH2042 3.0 W
Symbolic Computation
Laboratory II**

1998/99 CALENDAR COPY:*Advanced
symbolic computing with Maple. Topics from linear algebra,
differential equations, multivariate calculus, integral theorems, are
covered. Both mathematical understanding and applications are emphasized.
Enrolment is limited to about 30 per section.*

The general description of MATH2042 is similar to that of MATH2041. However, the topics covered will require greater mathematical maturity of the student. In particular, students will be required to have some knowledge of differential equations and multivariable calculus as well as the material required for MATH2041.

For a possible marking scheme and a list of projects see `
http://www.math.yorku.ca/Who/Faculty/Steprans/ Courses/2042.html`.

*Prerequisites: *AS/SC/MATH2041 3.0;

one of AS/SC/MATH2010 3.0, AS/SC/MATH2015 3.0,

AS/SC/AK/MATH2310 3.0; one of AS/SC/MATH1025 3.0,

AS/SC/MATH2021 3.0, AS/SC/AK/MATH2221 3.0.

*Corequisites: *AS/SC/AK/MATH2270 3.0;
one of

AS/SC/MATH2022 3.0,
AS/SC/AK/MATH2222 3.0.

*Degree credit exclusion: *AS/SC/MATH2040 6.0.

**Coordinator: ** J. Steprans

[See note to previous "Coordinator" entry.]

**AS/SC/MATH2090 3.0 W
Introduction to Mathematical Logic**

1998/99 CALENDAR COPY:*An
introduction to propositional logic; predicate logic, with an emphasis on
semantics; elements of axiomatic number theory. This course is intended for
Computer Science students and for Mathematics students who plan to do
further study in logic.*

*Important Note:* This year, this course will be in a somewhat
experimental, transitional stage. Two very different versions of it
wil be offered, corresponding to the fact that the backgrounds of
students taking it will fall into two very different categories.
**Sections N and P are intended for students who took MATH1090
BEFORE Fall term of 1998/99.** They are a degree credit exclusion with
sections A,B,D,E of MATH1090 F 1998/99. **Section M is intended for students who took the Computer
Science version of MATH1090 (sections A,B,D,E, Fall 1998/99).**
No other version of 1090, from the Fall or from an earlier year, is an
acceptable prerequisite for Section M. See Note 4 in the course
description of MATH1090 earlier in this minicalendar, for related
remarks.

*Prerequisite: *AS/SC/AK/MATH1090 3.0
or any 2000-level MATH course (without second digit 5)
or permission of the Course Coordinator.
Students who have not taken
MATH1090

are advised that familiarity with truth tables
will be assumed.

**Coordinators:** T.B.A.

**AS/SC/MATH2131 3.0
Introduction to Statistics II**

Pending approval by various higher bodies,
this course will be offered for the first time in 1999/2000. It
effectively replaces MATH1132 3.0. The prerequisite will be MATH2030
3.0.

**AS/SC/AK/MATH2221 3.0 FW
Linear Algebra with Applications I**

1998/99 CALENDAR COPY:*Systems
of linear equations, linear and affine subspaces of Euclidean
n-space, the Gauss-Jordan algorithm, matrices and matrix algebra,
determinants, vector space concepts for Euclidean n-space (linear dependence
and independence, basis, dimension, etc.), various applications.*

Linear algebra is a branch of mathematics which is particularly useful in other fields and in other branches of mathematics. Its frequent application in the engineering and physical sciences rivals that of calculus. Computer scientists and economists have long recognized its relevance to their discipline. Moreover, linear algebra is fundamental in the rapidly increasing quantification that is taking place in the management and social sciences. Finally, ideas of linear algebra are essential to the development of algebra, analysis, probability and statistics, and geometry.

This course and MATH2222 3.0 (see below) together provide a standard full-year introduction to linear algebra. While our focus will not be excessively theoretical, students will be introduced to proofs and expected to develop skills in understanding and applying concepts as well as results. Applications will be left mainly for MATH2222 3.0.

The text and grading scheme have not been determined as we go to press.

Note that MATH1540 3.0 may not be taken for credit by anyone who is taking, or anyone who has taken, MATH2221.

*Prerequisite: *OAC algebra or any university mathematics course.

*Degree credit exclusions: *AS/SC/MATH1025 3.0, AS/SC/

MATH2000 6.0,
AS/SC/MATH2021 3.0, AK/MATH2220 6.0.

**Coordinators:** Fall: R.G. Burns. Winter: I. Kleiner

**AS/SC/AK/MATH2222 3.0 FW
Linear Algebra with Applications II**

1998/99 CALENDAR COPY:*Linear
transformations and their representation by matrices, change of basis
and similarity, eigenvalues and eigenvectors, diagonalization, inner product
spaces, orthogonality, the Gram-Schmidt algorithm, least squares
approximations, abstract vector spaces, various applications.*

This course is a continuation of either MATH1025 3.0W or MATH 2221 3.0, and requires knowledge of the topics discussed in those courses.

The text for the *Fall* offering of this course will match the text for
MATH2221 from the Winter term of 1997/98.

The text for Winter term, and the grading schemes for both terms, have not been determined as we go to press.

*Prerequisite: *AS/SC/MATH1025 3.0

or AS/SC/AK/MATH2221 3.0.

*Degree credit exclusions: *AS/SC/MATH2000 6.0,

AS/SC/MATH2022 3.0,
AK/MATH2220 6.0.

**Coordinators:** Fall: S.O. Kochman. Winter: R.G. Burns

**AS/SC/AK/MATH2270 3.0 W
Differential Equations**

1998/99 CALENDAR COPY:*Introduction to differential equations, including a discussion
of the formation of mathematical models for real phenomena; solution
by special techniques; applications; linear equations; solutions in
series; other topics if time permits.*

[*Ed. note:* Repeated requests for input have elicited no reply
from Darewych or Laframboise, so we have just reproduced last year's
description here.]

This is a first course in differential equations. The curriculum will include first order ordinary differential equations of various types, including driven linear equations, exact equations, the Bernoulli equation and homogeneous equations. We will also study coupled systems of two first order equations (planar systems) with applications, including the pursuit problem, predator-prey models and other applications. Driven second order equations with constant coefficients will be studied by modeling small oscillations of a pendulum. We will solve these equations by introduction of the Green's kernel via variation of parameters, or by an application of Laplace transforms. Lastly, we will develop series techniques (including the Frobenius method) to find solutions of second order equations with non-constant coefficients.

*Prerequisites: *One of AS/SC/MATH2010 3.0,

AS/SC/MATH2015 3.0, or AS/SC/AK/MATH2310 3.0;

one of AS/SC/MATH1025 3.0,
AS/SC/MATH2021 3.0,

or AS/SC/AK/MATH2221 3.0.

**Coordinator: ** J. Laframboise

**AS/SC/MATH 2280 3.0 W
The Mathematical Theory
of Interest**

1998/99 CALENDAR COPY:*Topics
include measurement of interest, annuities, amortization of loans,
bonds, sinking funds and depreciation. The course is at a level which will
prepare students for the interest theory portion of the Society of Actuaries
examinations.*

This is a required course for students in the Actuarial

Stream, Mathematics for Commerce Honours Programme.

The text in 1997/98 was S. G. Kellison, *The Theory of Interest*
(Irwin Dorsey). It will most likely be the text for 1998/99, but
students should check with the instructor of the course.

*Prerequisite: *AS/SC/MATH1010 3.0 or

AS/SC/MATH1014 3.0 or
AS/SC/AK/MATH1310 3.0.

*Degree credit exclusion: *AS/AK/MATH2580 6.0.

**Coordinator: ** M. Abramson

**AS/SC/AK/MATH2310 3.0 FW
Calculus of Several
Variables with Applications**

1998/99 CALENDAR COPY:*Vector
functions, partial derivatives, gradient, multiple integrals, line
integrals, optimization, applications. Offered in both terms.*

This course is the sequel to MATH1300/1310. It is a required course in the ordinary programme in Mathematics, and in honours programmes in Mathematics for Commerce.

Just as MATH1300/1310 studied the calculus of functions of one variable, this course studies vectors and the calculus of functions of two or three variables. In addition to the above topics, we discuss tangent lines to space curves, tangent planes to surfaces, and change of variable in multiple integrals.

Students wishing to pursue calculus further may follow this course with MATH3010 3.0.

The central source for the course is the textbook. The lectures will not necessarily cover the same material as the text but will instead complement the material in the text. Reading the text and attending the lectures is essential for success in this course.

The text will be Salas and Hille, *Calculus: One and Several Variables*,
7th ed. (Wiley). (This text was used in 1997-98 in MATH1000/1010
and MATH1300/1310 and other courses. We will cover Chapters 12-16.)

The grade breakdown will be announced later.

*Prerequisite: *One of AS/SC/MATH1010 3.0,

AS/SC/MATH1014 3.0,
AS/SC/AK/MATH1310 3.0.

*Degree credit exclusions: *AS/SC/MATH2010 3.0,

AS/SC/MATH2015 3.0

**Coordinators:** Fall: S.O. Kochman. Winter: Y. Medvedev

**AS/SC/MATH 2320 3.0 F
Discrete Mathematical Structures**

1998/99 CALENDAR COPY:*This
course covers the algebraic and combinatorial structures that are
needed in computer science. Topics include set theory, functions, relations,
combinatorics, elements of graph theory, posets, lattices, Boolean algebras,
moniods, groups, morphisms, congruence relations. Intended primarily, but
not exclusively, for students in Computer Science.*

Based on consultation with the computer science department, we will emphasize topics which include: Big oh notation, complexity of formulae and algorithms, modular arithmetic, recursive definitions, general inductions, counting principles, recurrence relations and methods for solving them, trees and simple graph theory. The emphasis will include examples arising from algorithms and the ability to carry out analysis, problem solving, proofs and calculations which will be required in upper level courses.

The course does not require previous knowledge of computer science. A student of mathematics should enjoy this introduction to a variety of mathematical topics, many of which are not covered elsewhere.

This course is a prerequisite for COSC3101 3.0,

COSC3402 3.0,
COSC4101 3.0, COSC4111 3.0.

The text and grading scheme are not known as we go to press.

*Prerequisite: *AS/SC/AK/MATH1090 3.0 or any 2000-level MATH
course without second digit 5, or
permission of the Course Coordinator.

**Coordinator: ** T.B.A.

**AS/SC/AK/MATH2560 3.0 FW
Elementary Statistics I**

1998/99 CALENDAR COPY:*Displaying
and describing distributions, normal distibution. Relationships
between variables, regression and correlation. The need for design,
experimental design and sampling design. Sampling distributions, bias,
variability. Probability models, random variables, probability laws.*

Statistics is a collection of methods for observing and analyzing numerical data in order to make sensible decisions about them. In these courses the basic ideas of the analysis of data and of statistical inference will be introduced.

Little mathematical background is required; high school algebra is sufficient. Mathematical proofs will be minimal; reasoning and explanations will be based mostly on intuition, verbal arguments, figures, or numerical examples. Most of the examples will be taken from our daily life; many deal with the behavioural sciences, while others come from business, the life sciences, the physical sciences, and engineering.

Although students will be making some use of the computer to
calculate statistics, to create statistical plots, and to obtain
a better appreciation of statistical concepts, *no
previous
experience in computing is required*. Students will receive in
class all the necessary instruction about how to use the
statistical computer package Minitab.

Students who have taken MATH2560 3.0 will normally take MATH2570 3.0 in the second semester, where they will continue to investigate many basic statistical methods.

The text will be announced later.

The grading scheme has not yet been determined.

*Prerequisite: *Ontario Grade 12 Advanced Mathematics.

*Degree credit exclusions: *AS/SC/MATH1131 3.0,

SC/BIOL3080 3.0, SC/BIOL3090 3.0, AS/ECON2500 3.0,

AS/SC/GEOG2420 3.0, AS/SC/KINE2050 3.0,

AS/SC/PHED2050 3.0, AS/SC/PSYC2020 6.0,

AS/SC/PSYC2021 3.0, AS/SOCI3030 6.0,

AK/MATH1720 6.0, AK/MATH2430 6.0, AK/MATH2500 3.0,
AK/BIOL3080 6.0, AK/BIOL3080 3.0, AK/PSYC2510 3.0.

Not open to any student who
has successfully completed

AS/SC/MATH2030 6.0.

**Coordinators:** Fall: G. Denzel. Winter: D. Montgomery

**AS/SC/AK/MATH2570 3.0 W
Elementary Statistics II**

1998/99 CALENDAR COPY:*Binomial
distribution, sampling distribution of sample proportions and
means, central limit theorem. Confidence intervals, tests and decisions,
abuse of tests. Inference for a single mean, comparing two means, and for
spread. Contingency tables. Simple regression and basic analysis of
variance.*

See also the description for MATH2560 3.0. The text and grading scheme have not been decided upon as we go to press.

*Prerequisite: *AS/SC/AK/MATH2560 3.0.

*Degree credit exclusions: *AS/SC/MATH1132 3.0,

SC/BIOL3080 3.0, SC/BIOL3090 3.0, AS/ECON3210 3.0,

AS/ECON3500 3.0, AS/SC/GEOG2420 3.0,

AS/SC/PSYC2020 6.0, AS/SC/PSYC2022 3.0,

AS/SOCI3030 6.0, AK/MATH2430 6.0, AK/BIOL3080 6.0,

AK/BIOL3090 3.0, AK/PSYC3010 3.0, AK/PSYC3110 3.0.

Not open to any student who has successfully completed

AS/SC/MATH2030 6.0.

**Coordinator: ** G. Denzel

**AS/AK/MATH2580 6.0
Mathematics of Investment and
Actuarial Science**

1998/99 CALENDAR COPY:*Theory
of interest; annuities certain; amortization and sinking funds;
evaluation of bonds and other investments; depreciation, depletion and
capital cost; insurance, including mortality tables, principles of life
annuities, premiums and reserves.*

The first four-fifths of the course deal with most of the above topics, with applications to simple and general annuities, perpetuities, loan payments, capital budgeting, and internal rates of return. In the last few weeks of the course, the theory of interest is applied to life annuities and life insurance.

Students will use EXCEL, a spreadsheet available in the micro-computer
laboratory in the Steacie building. This spreadsheet operates on both
the Macintosh
and IBM-compatible platforms. EXCEL will be used to simplify and
illuminate
equation-solving, amortization of loans and mortgages, bond schedules,
depreciation tables, and mortality tables. *No
previous computer experience is assumed.* With the help of notes
and class instruction students will be introduced to the
spreadsheet and to its use in mathematics of finance. Students will receive
individual computer accounts on the ACADLABS server.

Each student will also need a hand-held calculator which has
*power* and *logarithm*
functions. Specifically, it must be able, given values
of *x* and *y* , to compute * x^y * (x to the power y

-- x positive, y real).

The course should be especially interesting to students of business and economics. The emphasis will be on practical problems. Although the mathematical background required is minimal, it is preferred that students will have taken one other mathematics course at university before taking this one.

The required text for the course will be P. Zima and R. Brown,
*Mathematics of Finance*, 4th edition
(McGraw-Hill Ryerson Ltd. 1993).
The manuals for EXCEL will be on reserve at the desk in Steacie Library.

Students who wish a more advanced treatment of the

material should *not* take this course, but enrol

instead in MATH2280 3.0. In particular, this includes:

1. Honours Mathematics for Commerce students in the Actuarial Stream. MATH2280 3.0 is a required course for this programme.

2. students who are contemplating a career in the actuarial
profession. They should take MATH2280 3.0, followed by

MATH3280 6.0.

In past years the final grade has been based on class testing (60%) and a final examination (40%). In 1998/99 there will likely be some spreadsheet-based homework assignments.

*Prerequisite: *One full university mathematics course.

*Degree credit exclusion: *AS/SC/MATH2280 3.0.

**Coordinator: ** Richard L.W. Brown

Contents Page, 1998/99 Minicalendar

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