# MATH Courses

# MATH 51 – Pre-University Algebra and Geometry

Topics covered in the course include operations with vectors, scalar multiplications dot and cross products, projections, equations of lines and planes, systems of equations, Gaussian elimination, operations with matrices, determinants, binomial theorem, proof by mathematical induction, complex numbers.

# MATH 52 – Pre-University Calculus

The concepts included are limits, derivatives, antiderivatives and definite integrals. These concepts will be applied to solve problems of rates of change, maximum and minimum, curve sketching and areas. The classes of functions used to develop these concepts and applications are: polynomial, rational, trigonometric, exponential and logarithmic.

# MATH 97 – Study Abroad

For studies at other universities under approved exchange agreements.

# MATH 103 – Introductory Algebra for Arts and Social Science

An introduction to applications of algebra to business, the behavioural sciences, and the social sciences. Topics will be chosen from linear equations, systems of linear equations, linear inequalities, functions, set theory, permutations and combinations, binomial theorem, probability theory. [Offered: F,W]

# MATH 104 – Introductory Calculus for Arts and Social Science

An introduction to applications of calculus in business, the behavioural sciences, and the social sciences. The models studied will involve polynomial, rational, exponential and logarithmic functions. The major concepts introduced to solve problems are rate of change, optimization, growth and decay, and integration. [Offered: F,W]

# MATH 106 – Applied Linear Algebra 1

Systems of linear equations. Matrix algebra. Determinants. Introduction to vector spaces. Applications. [Offered: F,W,S]

# MATH 109 – Mathematics for Accounting

Review and extension of differential calculus for functions of one variable. Multivariable differential calculus. Partial derivatives, the Chain Rule, maxima and minima and Lagrange multipliers. Introduction to matrix algebra.

# MATH 114 – Linear Algebra for Science

Vectors in 2- and 3-space and their geometry. Linear equations, matrices and determinants. Introduction to vector spaces. Eigenvalues and diagonalization. Applications. Complex numbers. [Offered: F]

# MATH 115 – Linear Algebra for Engineering

Linear equations, matrices and determinants. Introduction to vector spaces. Eigenvalues and diagonalization. Applications. Complex numbers. [Offered: F]

# MATH 116 – Calculus 1 for Engineering

Functions: review of polynomials, exponential, logarithmic, trigonometric. Operations on functions, curve sketching. Trigonometric identities, inverse functions. Derivatives, rules of differentiation. Mean Value Theorem, Newton's Method. Indeterminate forms and L'Hopital's rule, applications. Integrals, approximations, Riemann definite integral, Fundamental Theorems. Applications of the integral. [Offered: F]

# MATH 117 – Calculus 1 for Engineering

Functions of engineering importance; review of polynomial, exponential, and logarithmic functions; trigonometric functions and identities. Inverse functions (logarithmic and trigonometric). Limits and continuity. Derivatives, rules of differentiation; derivatives of elementary functions. Applications of the derivative, max-min problems, Mean Value Theorem. Antiderivatives, the Riemann definite integral, Fundamental Theorems. Methods of integration, approximation, applications, improper integrals. [Offered: F]

# MATH 118 – Calculus 2 for Engineering

Methods of integration: by parts, trigonometric substitutions, partial fractions; engineering applications, approximation of integrals, improper integrals. Linear and separable first order differential equations, applications. Parametric curves and polar coordinates, arc length and area. Infinite sequences and series, convergence tests, power series and applications. Taylor polynomials and series, Taylor's Remainder Theorem, applications. [Offered: W,S]

# MATH 119 – Calculus 2 for Engineering

Elementary approximation methods: interpolation; Taylor polynomials and remainder; Newton's method, Landau order symbol, applications. Infinite series: Taylor series and Taylor's Remainder Theorem, geometric series, convergence test, power series, applications. Functions of several variables: partial derivatives, linear approximation and differential, gradient and directional derivative, optimization and Lagrange multipliers. Vector-valued functions: parametric representation of curves, tangent and normal vectors, line integrals and applications. [Offered: W,S]

# MATH 124 – Calculus and Vector Algebra for Kinesiology

Review of trigonometry and basic algebra. Introduction to vectors in 2- and 3-space: sums, addition, dot products, cross products and angles between vectors. Solving linear systems in two and three variables. Functions of a real variable: powers, rational functions, trigonometric, exponential and logarithmic functions, their properties. Intuitive discussion of limits and continuity. Derivatives of elementary functions, derivative rules; applications to curve sketching, optimization. Relationships between distance, velocity and acceleration. The definite integral, antiderivatives, the Fundamental Theorem of Calculus; change of variable and integration by parts; applications to area, centre of mass. [Offered: F]

# MATH 127 – Calculus 1 for the Sciences

Functions of a real variable: powers, rational functions, trigonometric, exponential and logarithmic functions, their properties and inverses. Intuitive discussion of limits and continuity. Definition and interpretation of the derivative, derivatives of elementary functions, derivative rules and applications. Riemann sums and other approximations to the definite integral. Fundamental Theorems and antiderivatives; change of variable. Applications to area, rates, average value. [Offered: F,W,S]

# MATH 128 – Calculus 2 for the Sciences

Transforming and evaluating integrals; application to volumes and arc length; improper integrals. Separable and linear first order differential equations and applications. Introduction to sequences. Convergence of series; Taylor polynomials, Taylor's Remainder Theorem, Taylor series and applications. Parametric/vector representation of curves; particle motion and arc length. Polar coordinates in the plane. [Offered: F,W,S]

# MATH 135 – Algebra for Honours Mathematics

An introduction to the language of mathematics and proof techniques through a study of the basic algebraic systems of mathematics: the integers, the integers modulo n, the rational numbers, the real numbers, the complex numbers and polynomials. [Offered: F,W,S]

# MATH 136 – Linear Algebra 1 for Honours Mathematics

Systems of linear equations, matrix algebra, elementary matrices, computational issues. Real n-space, vector spaces and subspaces, basis and dimension, rank of a matrix, linear transformations and matrix representations. Determinants, eigenvalues and diagonalization, applications. [Offered: F,W,S]

# MATH 137 – Calculus 1 for Honours Mathematics

Absolute values and inequalities. Sequences and their limits. Introduction to series. Limits of functions and continuity. The Intermediate Value Theorem and approximate solutions to equations. Derivatives, linear approximation and Newton's method. The Mean Value Theorem and error bounds. Applications of the Mean Value Theorem, Taylor polynomials and Taylor's Theorem, Big-O Notation, Suitable topics are illustrated using computer software. [Offered: F,W,S]

# MATH 138 – Calculus 2 For Honours Mathematics

Introduction to the Riemann Integral and approximations. Antiderivatives and the Fundamental Theorem of Calculus. Change of variables, Methods of integration. Applications of the integral. Improper integrals. Linear and separable differential equations and applications. Tests for convergence for series. Binomial Series, Functions defined as power series and Taylor series. Vector (parametric) curves in R2. Suitable topics are illustrated using computer software. [Offered: F,W,S]

# MATH 145 – Algebra (Advanced Level)

MATH 145 is an advanced-level version of MATH 135. [Offered: F]

# MATH 146 – Linear Algebra 1 (Advanced level)

MATH 146 is an advanced-level version of MATH 136. [Offered: W]

# MATH 147 – Calculus 1 (Advanced Level)

MATH 147 is an advanced-level version of MATH 137. [Offered: F]

# MATH 148 – Calculus 2 (Advanced Level)

MATH 148 is an advanced-level version of MATH 138. [Offered: W]

# MATH 199 – Mathematical Discovery and Invention

A course in problem solving in which intriguing and difficult problems are solved. Problems are taken mainly from the elementary parts of applied mathematics, computer science, statistics and actuarial science, pure mathematics, and combinatorics and optimization. Material relevant to the problems is taught in depth.

# MATH 207 – Calculus 3 (Non-Specialist Level)

Multivariable functions and partial derivatives. Gradients. Optimization including Lagrange multipliers. Polar coordinates. Multiple integrals. Surface integrals on spheres and cylinders. Introduction to Fourier Series. [Offered: F,W,S]

# MATH 211 – Advanced Calculus 1 for Electrical and Computer Engineers

Fourier series. Ordinary differential equations. Laplace transform. Applications to linear electrical systems. [Offered: F,W]

# MATH 211N – Advanced Calculus 1 for Nanotechnology Engineering

Ordinary differential equations with constant coefficients. Boundary value problems and applications to quantum mechanics. Laplace transforms, Fourier series and applications. Numerical solution of ordinary differential equations. [Offered: F]

# MATH 212 – Advanced Calculus 2 for Electrical Engineers

Triple integrals, cylindrical and spherical polar coordinates. Divergence and curl, applications. Surface integrals, Green's, Gauss' and Stokes' theorems, applications. Complex functions, analytic functions, contour integrals, Cauchy's integral formula, Laurent series, residues. [Offered: F,S]

# MATH 212N – Advanced Calculus 2 for Nanotechnology Engineering

Gradient, Divergence and Curl: Applications. Line and Surface Integrals. Green's, Gauss', and Stokes' Theorems: Applications to electromagnetism and fluid mechanics. Numerical solution of partial differential equations. [Offered: S]

# MATH 213 – Advanced Mathematics for Software Engineers

Fourier series. Differential equations. Laplace transforms. Applications to circuit analysis. [Offered: S]

# MATH 215 – Linear Algebra for Engineering

Systems of linear equations; their representation with matrices and vectors; their generalization to linear transformations on abstract vector spaces; and the description of these linear transformations through quantitative characteristics such as the determinant, the characteristic polynomial, eigenvalues and eigenvectors, the rank, and singular values. [Offered F,W]

# MATH 217 – Calculus 3 for Chemical Engineering

Curves and surfaces in R3. Multivariable functions, partial derivatives, the chain rule, gradients. Optimization, Lagrange Multipliers. Double and triple integrals, change of variable. Vector fields, divergence and curl. Vector integral calculus: Green's theorem, the Divergence theorem and Stokes' theorem. Applications in engineering are emphasized. [Offered: F,W]

# MATH 218 – Differential Equations for Engineers

First order equations, second order linear equations with constant coefficients, series solutions, the Laplace transform method, systems of linear differential equations. Applications in engineering are emphasized. [Offered: F,S]

# MATH 225 – Applied Linear Algebra 2

Vector spaces. Linear transformations and matrices. Inner products. Eigenvalues and eigenvectors. Diagonalization. Applications. [Offered: F,S]

# MATH 227 – Calculus 3 for Honours Physics

Directional derivative and the chain rule for multivariable functions. Optimization, Lagrange multipliers. Double and triple integrals on simple domains; transformations and Jacobians; change of variable in multiple integrals. Vector fields, divergence and curl. Vector integral calculus: Line and surface integrals, Green's Theorem, Stokes' Theorem, Gauss' Theorem, conservative vector fields. [Offered: F]

# MATH 228 – Differential Equations for Physics and Chemistry

First-order equations, second-order linear equations with constant coefficients, series solutions and special functions, the Laplace transform method. Applications in physics and chemistry are emphasized. [Offered: F,W]

# MATH 229 – Introduction to Combinatorics (Non-Specialist Level)

Introduction to graph theory: colourings, connectivity, Eulerian tours, planarity. Introduction to combinatorial analysis: elementary counting, generating series, binary strings. [Offered: F]

# MATH 235 – Linear Algebra 2 for Honours Mathematics

Orthogonal and unitary matrices and transformations. Orthogonal projections, Gram-Schmidt procedure, best approximations, least-squares. Inner products, angles and orthogonality, orthogonal diagonalization, singular value decomposition, applications.

# MATH 237 – Calculus 3 for Honours Mathematics

Calculus of functions of several variables. Limits, continuity, differentiability, the chain rule. The gradient vector and the directional derivative. Taylor's formula. Optimization problems. Mappings and the Jacobian. Multiple integrals in various co-ordinate systems.

# MATH 239 – Introduction to Combinatorics

Introduction to graph theory: colourings, matchings, connectivity, planarity. Introduction to combinatorial analysis: generating series, recurrence relations, binary strings, plane trees.

# MATH 245 – Linear Algebra 2 (Advanced Level)

MATH 245 is an advanced-level version of MATH 235. [Offered: F,S]

# MATH 247 – Calculus 3 (Advanced Level)

Topology of real n-dimensional space: completeness, closed and open sets, connectivity, compact sets, continuity, uniform continuity. Differential calculus on multivariable functions: partial differentiability, differentiability, chain rule, Taylor polynomials, extreme value problems. Riemann integration: Jordan content, integrability criteria, Fubini's theorem, change of variables. Local properties of continuously differentiable functions: open mapping theorem, inverse function theorem, implicit function theorem. [Offered: F,W,S]

# MATH 249 – Introduction to Combinatorics (Advanced Level)

MATH 249 is an advanced-level version of MATH 239. [Offered: F,W]

# MATH 455W – Partial Differential Equations (WLU)

# MATH 475W – Ring And Field Theory (WLU)

# MATH 600 – Introduction to Mathematical Software for Teachers

This course exposes students to the technical tools that professional mathematicians use. The software presented in the course will enhance each student's communication, presentation, visualization, and problem-solving skills. The course will also take a brief look at the history of mathematical communication and its impact on the development of the subject.

# MATH 630 – Foundations of Probability

This course will explore the basic properties of probability focusing on both discrete and continuous random variables. Topics include: Laws of probability, discrete and continuous random variables, probability distributions, mean, variance, generating functions, Markov chains, problem solving, history of probability.

# MATH 631 – Statistics for Teachers

This course discusses some of the mathematical and scientific aspects of empirical, or data-based, problem solving. Topics will include methods for the design of experiments and surveys, and the analysis of data using statistical models. Examples will illustrate the application of these methods to data-based problems in science, health, business and industry.

# MATH 636 – Linear Algebra for Teachers

This course explores the foundations of linear algebra and some of its applications. An emphasis will be placed on the development of mathematical thinking and the importance of proof in mathematical teaching. Topics include: matrices, linear mappings, vector spaces, determinants, diagonalization, inner products, the Fundamental Theorem of Linear Algebra, and the method of least squares.

# MATH 640 – Number Theory for Teachers

This course explores the many fascinating properties of the natural numbers. Topics include: the Euclidean algorithm, congruences and modular arithmetic, primitive roots and quadratic residues, sums of squares, multiplicative functions, continued fractions and Diophantine equations, and rational approximations to real numbers.

# MATH 641 – Algorithm Design and Analysis

This course explores the intersection between mathematics and computer science by examining general methods for solving problems efficiently. We will study a variety of algorithm design techniques for problems in diverse application areas and use mathematical models to compare algorithms. Students will be introduced to "big ideas" from computer science, such as the generalization and reuse of previous solutions, and the notion of limits to the power of computation.

# MATH 642 – Introduction to Computer Science: A Mathematical Perspective

The connections between mathematics and computer science are varied and deep. This course will introduce students to foundational ideas in computer science and their relationship to foundational ideas in mathematics through the use of a functional programming language designed for education. No prior experience with programming is required, though students with exposure to popular programming languages will also benefit from this alternate approach.

# MATH 643 – Theory of Computation

This course will use mathematics to study the foundations of problem solving and computation. A decision problem (answering "yes" or "no") can be characterized as a set of strings of characters encoding inputs that yield "yes" answers, and a computer can be characterized as a simple mathematical machine model that processes strings of characters. By studying properties of classes of problems of increasing complexity, we will be able to establish relationships among classes, membership in classes, and hard problems for classes. The course will culminate in the examination of classes and models that correspond to the power of computers, and the demonstration that there exist problems that cannot be solved.

# MATH 647 – Foundations of Calculus I

This course will explore the foundations of differential calculus, the role of rigor in mathematics, and the use of sophisticated mathematical software. Topics include: A brief primer on logic and proof, axiom of choice and other ideas from set theory, convergence of sequences and the various forms of the completeness axiom for R, detailed study of limits, continuity and the Intermediate Value Theorem, fundamentals of differentiation and the importance of linear approximation, role of the Mean Value Theorem, the nature and existence of extrema, Taylor's Theorem and polynomial approximation, MAPLE as a tool for discovery.

# MATH 648 – Foundations of Calculus II

This course explores the foundations of integral calculus and the use of series in approximating the basic functions of mathematics. Topics include: Understanding the Riemann Integral and its flaws, the idea of Lebesque, the geometric meaning of the Riemann-Stieltjes integral, the Fundamental Theorem of Calculus, numerical integration, numerical series, uniform convergence of functions and the extraordinary nature of power series, Fourier Series.

# MATH 650 – Mathematical Modeling with Differential Equations

Solving and interpreting differential and difference equations motivated by a variety of systems from the physical and social sciences. Analytical solutions of standard linear and non-linear equations of first and second order; phase portrait analysis; linearization of non-linear systems in the plane. Numerical and graphical solutions using mathematical software.

# MATH 660 – Explorations in Geometry

This course is designed to allow the student to discover fundamental facts about geometry through the interactive use of mathematical software. Possible topics include: An introduction to affine, projective and non-Euclidean geometry, conic sections in projective geometry, inversion in circles, the Theorems of Desargues, Pappas and Pascal.

# MATH 661 – Problem Solving and Proof in Geometry

This course explores the Euclidean geometry of triangles and circles from elementary to advanced settings. The course also briefly discusses three-dimensional geometry. An emphasis is placed on proof, on problem solving, on problem creation in a geometric context, and on aspects of teaching geometry. Where possible, problems that combine multiple areas of mathematics are used.

# MATH 670 – Mathematical Connections: Real World Problems in Mathematics I

This course is intended to give the student insight to an important area of mathematics and how it connects with problems in the real world. Topics: The course is one of four similar courses that will consist of either one six week module, or two three week modules each introducing a separate though possibly related area of mathematics. The emphasis will be on how the mathematics is used in a real world context.

# MATH 671 – Mathematical Connections: Real World Problems in Mathematics II

This course is intended to give the student insight to an important area of mathematics and how it connects with problems in the real world. Topics: The course is one of four similar courses that will consist of either one six week module, or two three week modules each introducing a separate though possibly related area of mathematics. The emphasis will be on how the mathematics is used in a real world context.

# MATH 672 – Mathematical Connections: Real World Problems in Mathematics III

This course is intended to give the student insight to an important area of mathematics and how it connects with problems in the real world. Topics: The course is one of four similar courses that will consist of either one six week module, or two three week modules each introducing a separate though possibly related area of mathematics. The emphasis will be on how the mathematics is used in a real world context.

# MATH 673 – Mathematical Connections: Real World Problems in Mathematics IV

# MATH 674 – Special Topics in Mathematical Connections

This course is intended to give the student insight to an important area of mathematics and how it connects with problems in the real world. The course will consist of either a one six week module, or two three week modules each introducing a separate though possibly related area of mathematics. The emphasis will be on how the mathematics is used in a real world context.

# MATH 680 – History of Mathematics

We explore the who, where, when and why of some of the most important ideas in mathematics. Topics include: William T. Tutte and Decryption, Euclid and the Delian Problem, Archimedes and his estimate of Pi, Al Khwarizmi and Islamic mathematics, Durer and the Renaissance, Descartes and Analytic Geometry, and Kepler and Planetary Motion.

# MATH 681 – Problem Solving and Mathematical Discovery

This course aims to develop the student's mathematical problem solving ability. Common heuristics such as problem modification, patterning, contradiction arguments and exploiting symmetry will be examined. A wide range of challenging problems from various branches of mathematics will provide the medium through which these important principles and broad strategies are experienced.

# MATH 690 – Summer Conference for Teachers of Mathematics

This intense 3-day workshop focuses on the integration of problem solving technology into the curriculum and enrichment activities. The Workshop is suitable for teachers from all over the world.

# MATH 692 – Reading, Writing and Discovering Proofs

Objectives: To develop the vocabulary, techniques and analytical skills associated with reading and writing proofs, and to gain practice in formulating conjectures and discovering proofs. Emphasis will be placed on understanding logical structures, recognition and command over common proof techniques, and precision in language. Topics Include: rules of formal logic, truth tables, role of definitions, implications, sets, existential and universal quantifiers, negation and counter-example, proofs by contradiction, proofs using the contrapositive, proofs of uniqueness and induction.

# MATH 698 – Reading Course in Mathematics for Teachers

Students will undertake a reading, research and writing project on a mathematical topic of interest to teachers.

# MATH 699 – Master of Mathematics for Teachers Capstone

The capstone course is designed to give students an opportunity to showcase the knowledge they have gained and to provide a forum for bringing that knowledge into their own classroom. As part of this course students will design a mini-course on an approved subject in mathematics.