Second Year Courses

Offered 2015-2016

MATH 2080Y – Mathematics for Teacher Education

MATH 2110H – Calculus III: Calculus of Several Variables

MATH 2120H – Calculus IV: Vector Calculus

MATH-PHYS 2150H – Ordinary Differential Equations

MATH 2200H – Mathematical Reasoning

MATH 2260H – Geometry I: Euclidean Geometry

MATH 2350H – Linear Algebra II: Vector Spaces

MATH 2560H –Statistics I: Introductions and Applications

MATH-COIS 2600H – Discrete Structures


Math 2080Y - Mathematics for Teacher Education

 A course in mathematics and mathematical thinking for prospective elementary school teachers. Number systems and counting, graphs and networks, symmetry and patterns, mathematics in nature and art, probability and statistics, measurement and growth. Does not satisfy the Mathematics requirement for a Bachelor of Science degree.

  • Note: Instructor's approval required; enrolment limited.
  • Prerequisite: Normally open only to students in the Concurrent
    Education program or who are pursuing the Emphasis in Teacher Education.
  • Excludes: MATH 280 and any MATH course, or its equivalent, which counts toward a major or minor in Mathematics.

Math 2110H - Calculus III: Calculus of Several Variables

 Multivariable functions, curves, and surfaces in two and three dimensions. Partial differentiation and applications. Multiple integrals. Prerequisite: Excludes MATH 200, 201H.

  • Prerequisite: 60% or higher in MATH 1350H and in one of 1120H or 1100Y or 1101Y.
  • Excludes: MATH 200, 201H.

Math 2120H - Calculus IV: Vector Calculus

 Parametric curves and surfaces, vector functions and fields. Line Integrals, Green’s Theorem. Surface integrals, curl and divergence, Stokes’ and Divergence Theorems.

  • Prerequisite: 60% or higher in MATH 2110H.
  • Excludes: MATH 200, 202H, MATH 2110H (201H).

Math-Phys 2150H - Ordinary Differential Equations

Synopsis::  Analytical and graphical solutions are studied for first-order, higher-order and systems of first-order equations. Applications are shown by linear and nonlinear Mathematical models in Physics, Biology and other areas. Laplace transform and its applications are also introduced. (Time permitting, power series solutions and/or numerical methods may also be covered.)

Overview: Ordinary differential equations (ODEs) model physical systems which evolve continuously in time. For example, suppose the state of the system is described by a single variable x(t), and satisfies the equation:


d x(t)
----- = -x(t).
d t
If x(0)=5, then the unique solution to this equation is the curve x(t) = 5 e-t. This curve starts at 5, and asymptotically approaches 0. We say that 0 is an equilibrium state for the system.

Suppose the state of the system at time t is given by a real vector x(t) in RN. Then an ODE for this system has the form:


d x
----- = V[x(t)].
d t
This says that the velocity of the system (namely dx/dt) is a determined entirely by it's state (namely x), via some function V:RN ---->RN.

ODEs are ubiquitous in applied mathematics. For example:


  • In physics: ODEs model trajectories in classical and relativistic mechanics.
  • In chemistry: ODEs describe the reaction kinetics of chemical systems.
  • In biology: ODEs describe the evolving populations of interacting species, fluctuating endocrine levels in the body, or neural activity in the brain.
  • In economics: ODEs describe business cycles.

Given an ODE, we can ask four questions:

  1. Do solutions exist? In other words, given an initial state x0, is there a smooth curve x(t) satisfying x(0)=x0 and the ODE?
  2. Is this solution unique?
  3. What is an explicit formula describing the solution?
  4. What is the long-term qualitative behaviour of the system?
  • Prerequisite: 60% or higher in MATH 1120H or 1100Y or 1101Y.
  • Recommended: MATH 1350H (135H).
  • Excludes: MATH-PHYS 205H.

Math-Cois 2180H - Introduction to Numerical and Computational Methods

 Error analysis, nonlinear equations, linear systems, interpolation methods, numerical differentiation and integration and initial value problems.

  • Prerequisite: 60% or higher in one of MATH 1005H or 1120H or 1100Y or 1101Y.
  • Excludes: MATH 207H, MATH-COSC 203H.

Math 2200H - Mathematical Reasoning

 This course is intended for mathematics majors in the first or second year of their program, and introduces the essential concepts, methods, paradigms and abstractions of higher mathematics. It is recommended as a pre/corequisite for upper level pure math courses such as MATH 3720H[302H] , 307H, 309H, 310H, 320H, 321H, 332H, 336H, 407H, 409H, 410H, 415H, 431H, 432H, 433H, 435H, 437H, 416H, 461H, 462H, 471H, and 472H. We will cover most of the following topics:

  • Proofs and logic: Modus ponens; proof by cases; proof by contradiction (eg: Euclid's proof that there are infinitely many prime numbers; Cantor's diagonal argument). Proof by induction (e.g. the Euclidean algorithm; Lamé's theorem.)
  • Sets and functions: Sets and subsets; union and intersection; de Morgan's laws. Functions: injective, surjective, and bijective. Inverse images.
  • Combinatorics: Counting arguments; the Pigeonhole Principle; factorials, binomial coefficients, Pascal's formula; the binomial theorem. Multinomial coefficients and the multinomial theorem.
  • Transfinite Arithmetic: Cantor's definition of cardinality; The sets N, Z, N2, and Q all have cardinality Aleph0. Cantor's diagonal argument, showing that R has cardinality c > Aleph0.
  • Complex numbers: The complex plane in Cartesian and polar coordinates. Addition, multiplication, and the exponential map, and their geometric interpretations. de Moivre's formulae.
  • Basic Number Theory:
    • Divisibility; greatest common divisors, lowest common multiples. The Euclidean algorithm. Fibonacci numbers and Lamé's theorem.
    • Linear diophantine equations. Brahmagupta's theorem.
    • Prime numbers and prime factorization. The Fundamental Theorem of Arithmetic.
    • Modular arithmetic: Modular addition and multiplication; the concept of equivalence class, with congruence classes as the prototypical example. Linear congruence equations. Fermat's little theorem.
  • Possible additional topics: We may have time for one or two of the following topics:
    • Symmetry and Transformation groups: Symmetries of plane figures. Transformation groups. Examples: dihedral group, matrix groups, permutation groups. Subgroups, group homomorphisms and group isomorphisms.
    • Introduction to topology: Curves, surfaces, and manifolds. Homeomorphism and homotopy. The Euler-Poincaré invariant and the fundamental group.
  • Prerequisite or Corequisite: 60% or higher in MATH 1350H and in one of MATH 1120H or 1100Y or 1101Y.
  • Excludes: MATH 220H.

Math 2260H - Geometry I: Euclidean Geometry

 Geometry is the oldest part of mathematics; its origins are lost in antiquity. Classical Euclidean geometry applies logical deduction to discover the relationships of lines, points, and circles in the plane. For more than two thousand years, it has been the ideal of clarity and elegance to which all other mathematics aspires. In this course, we will cover some subset of the following topics:

  • Constructions using compass & straight-edge. Nonconstructability.
  • Triangles:
    • Similarity & congruence.
    • Ceva's theorem and corollaries.
  • Symmetry:
    • Isometries; Kleinian geometry.
    • Symmetric planar figures.
    • Friezes, tilings, and crystal structures.
    • Symmetric polyhedra (Platonic & Archimedean).
  • Isaac Newton's geometric development of calculus in the Principia.
  • (time permitting) Introduction to non-Euclidean geometry:
    • The Parallel Postulate.
    • Spherical geometry.
    • Hyperbolic geometry.
    • Toroidal geometry.

Recommended for Education students.

  • Prerequisite: 60% or higher in one of MATH 1005H, 1120H, 1100Y, 1101Y, or 1350H.
  • Excludes: MATH 226H.


Math 2350H - Linear Algebra II: Vector Spaces

Vector spaces, basis and dimension, inner product spaces, orthogonality, linear transformations, diagonalization, determinants, eigenvalues, quadratic forms, least squares, the singular value decomposition.

  • Prerequisite: 60% or higher in MATH 1350H.
  • Excludes: MATH 235H.

Math 2560H - Statistics I: Introduction and Applications

 An introduction to applied statistical methods. Graphical and numerical presentation of data, probability distributions and central limit theorem, methods of point estimation, confidence intervals, hypotheses testing,
comparative inferences, nonparametric methods. Assumes a background in calculus.

  • Prerequisite: 60% or higher in MATH 1550H. Assumes a background in probability and calculus.
  • Excludes: MATH 355, 256H.

Math-Cois 2600H - Discrete Structures

 Mathematics related to computer science, including sets and relations, counting techniques and recursive relations, trees and networks. Applications to analysis of algorithms, data structure, and optimization problems.

  • Prerequisite: 60% or higher in MATH 1350H and in one of MATH 1120H or 1100Y or 1101Y; or 60% or higher in COIS 1020H, MATH 1005H, and 1350H.
  • Excludes: MATH-COSC 260.