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# 2000-level Courses

This is the first in a three-course sequence in Analysis (to be followed by MATH 3001 and MATH 4010) for Honours stream Mathematics majors. The course proves that the real numbers, seen as infinite sequences of digits, satisfy various compactness properties. This is followed by a study of limits of sequences. The convergence of sequences is then used to define the continuity of functions on the real line. Differentiation and Riemann integration are then developed, filling in the details of proofs that are not usually provided in first year calculus courses.

Prerequisites: SC/MATH 1200 3.00, SC/MATH 1310 3.00.

Course credit exclusion: SC/MATH 3110 3.00, GL/MATH 3320 3.00.

Note: This course is a prerequisite for SC/MATH 3001 3.00.

This is a one-semester course in theory and applications of multivariable and integral calculus designed for students in engineering and the physical sciences. Topics covered in this course include functions of several variables; arclength and curvature; partial derivatives; grad, div, curl and Laplacian operators; extrema and Taylor series for multivariate functions; double and triple integrals in various coordinate systems; line and surface integrals; theorems of Green, Gauss and Stokes. As this is an extension from techniques in single-variable calculus, understanding of prerequisite material is critical for success.

Prerequisite: One of SC/MATH 1010 3.00, SC/MATH 1014 3.00, SC/MATH 1310 3.00; or SC/MATH 1507 3.00 plus permission of the course coordinator.

Course credit exclusions: SC/MATH 2010 3.00, SC/MATH 2310 3.00, GL/MATH 2670 3.00, GL/MODR 2670 3.00, GL/MATH 3200 3.00.

Note: This course is a prerequisite for SC/MATH 2131 3.00, SC/MATH 2270 3.00, SC/MATH 2271 3.00, SC/MATH 3271 3.00, SC/MATH 3410 3.00, SC/MATH 4120 3.00, SC/MATH 4143 3.00, SC/MATH 4171 3.00, SC/MATH 4172 3.00, SC/MATH 4271 3.00 and SC/MATH 4830 3.00.

This course is the continuation of Math 1021 (Linear Algebra I) and will pursue linear algebra from a more abstract point-of-view. In this course, students will learn about general vector spaces, linear transformations, inner product spaces, orthogonality, spectral theory, and canonical forms of matrices. Time permitting additional topics and applications will be discussed. MATH 2022 is intended to enhance students' critical thinking, reasoning, and communication skills. As such, written assignments with questions geared towards mathematics majors are a part of this course.

Prerequisite: one of SC/MATH 1021 3.00, SC/MATH 2021 3.00, GL/MATH/MODR 2650 3.00 or permission of the course coordinator.

Course credit exclusions: SC/MATH 2222 3.00, GL/MATH/MODR 2660 3.00.

Note: This course is a prerequisite for SC/MATH 2271 3.00, SC/MATH 3010 3.00, SC/MATH 3021 3.00, SC/MATH 3050 6.00, SC/MATH 3052 6.00, SC/MATH 3090 3.00, SC/MATH 4160 3.00 and SC/MATH 4630 3.00.

This course provides an introduction to the theory of probability. It covers the mathematics used to calculate proba­bilities and expectations and discusses how random variables can be used to pose and answer interesting problems arising in nature. It is required for most programs in Math­ematics and Statistics, or in Computer Science. Subsequent courses that use the material covered include mathematical statistics, operations research, mathematical finance, stochastic pro­cesses, as well as more advanced courses in probability.

Prerequisite: One of SC/MATH 1014 3.00, SC/MATH 1310 3.00, SC/ISCI 1402 3.00, SC/ISCI 1410 6.00.

Note: This course is a prerequisite for SC/MATH 2131 3.00, SC/MATH 2281 3.00, SC/MATH 3090 3.00, SC/MATH 4143 3.00, SC/MATH 4172 3.00, SC/MATH 4430 3.00, SC/MATH 4431 3.00 and SC/MATH 4931 3.00.

Many of the technological advances that come from scientific innovation depend on efficient means of computation and analysis of large amounts of data. Before digital computers became widely available, these sorts of computations were largely done by hand or with the aid of mechanical calculation tools or tables. However, for every computation tool that we have available, there are always mathematical problems that are beyond the limits of our computational power. For example, the ability to factor an 800 digit integer which is the product of two primes or find the determinant of a large matrix is out of the reach of our current computational tools. This course will use the program Maple to answer numerical and discrete computation questions which would otherwise be too difficult to do by hand or the use of a simple calculator. Maple is an example of a Computer Algebra System (CAS) but other examples such as sage, Mathematica and Matlab are similarly capable of doing a wide range of computations and other many specialized programs (such as R, GAP and Macaulay) are particularly efficient at certain types of computations.

Prerequisites: LE/EECS 1560 3.00 or equivalent computing experience; SC/MATH 1014 3.00 or SC/MATH 1310 3.00 or SC/ISCI 1401 3.00 or SC/ISCI 1410 6.00.

Note: This course is a prerequisite for SC/MATH 3090 3.00 and SC/MATH 4143 3.00.

This course serves as an introduction to mathematical statistics and is devoted to the study of the basic probability tools needed in the theory of statistical inference. Topics include joint distributions, multivariate change of variables formula, conditional and marginal distributions, conditional expectation, covariance and correlation, and moment generating functions. Dis­tri­butional results including those associated with normally distributed observations are examined. The course ends with a look at some statistical applications such as ANOVA or linear regression (time permitting).

The topics considered in this course require a solid knowledge of univariate and multivariate calculus.

Prerequisites: SC/MATH 1131 3.00; SC/MATH 2030 3.00; SC/MATH 2015 3.00 or SC/MATH 2310 3.00.

Note: This course is a prerequisite for SC/MATH 3131 3.00, SC/MATH 3280 3.00, SC/MATH 3430 3.00, SC/MATH 4280 3.00 and SC/MATH 4281 3.00.

This class will be offered for students did not take Math 1200 in their first two years of study. One goal of the course is to develop communication and analytic skills that will allow students to begin to solve problems where there is no obvious method of solution. This course focuses on developing stronger communication skills, both written and oral, while presenting a convincing argument in mathematic. This involves using clearly defined language, logic, and symbolic manipulation in a way that conveys to the reader that a mathematical statement is true. The topics in this course vary widely, but using definitions and theorems in proofs is a central theme, and material will come from different sources such as, but not limited to, algebra, graph theory, number theory, game theory, analysis, and topology.

Prerequisites: SC/MATH 1300 3.00 and SC/MATH 1310 3.00, or SC/ISCI 1401 3.00 and SC/ISCI 1402 3.00, or SC/ISCI 1410 6.00; SC/MATH 1021 3.00 or equivalents; taking or has taken a math course at the 3000 level or higher.

Course credit exclusion: SC/MATH 1200 3.00

Note: This course is a prerequisite for SC/MATH 3141 3.00.

Differential equations have played a central role in mathematics and its applications for the past three hundred years. Their importance in ap­pli­ca­tions stems from the interpretation of the derivative as a rate of change, a familiar example being velocity. Many of the fundamental laws of physical science are best formulated as differential equa­tions. In other areas, too, such as biology and ec­onomics, which involve the study of growth and change, such equations are of fundamental im­portance. In this course we will study some important types of linear differential equations and their solutions. Topics will include first-order (differential) equations; homogeneous second and higher order equations with constant coefficients; the particular solution of in­homogeneous second-order equations; first-order linear systems, solutions and phase plane; series-form solutions of equations with variable coefficients; solutions by use of Laplace transforms. Some nonlinear systems will be explored using linearization and phase portrait analysis.

Prerequisites: One of SC/MATH 2015 3.00 or SC/MATH 2310 3.00; one of SC/MATH 1021 3.00, SC/MATH 1025 3.00, or SC/MATH 2221 3.00.

Course credit exclusion: SC/MATH 2271 3.00, GL/MATH 3400 3.00.

Note: This course is a prerequisite for SC/MATH 3271 3.00, SC/MATH 4120 3.00, SC/MATH 4141 3.00 and SC/MATH 4271 3.00.

This course gives an overview of differential equations for students in science and engineering. The emphasis is on ordinary differential equations, and the classical methods of solutions for a variety of types of equations are covered. General first order equations, as well as linear second order equations, are discussed, both in terms of general theory and particular solution techniques. Series solutions for second order equations are presented. Methods of solution for second order linear equations are extended to higher order equations. Boundary value problems for partial differential equations are presented, with the main solution technique being separation of variables and Fourier series.

Prerequisites: One of SC/MATH 2015 3.00, SC/MATH 2310 3.00 or equivalent; one of SC/MATH 1025 3.00, SC/MATH 2022 3.00, SC/MATH 2222 3.00 or equivalent.

Course Credit Exclusions: SC/MATH 2270 3.00, GL/MATH 3400 3.00.

Note: This course is a prerequisite for SC/MATH 4120 3.00 and SC/MATH 4271 3.00.

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.

Prerequisite: SC/MATH 1014 3.00 or SC/MATH 1310 3.00 or SC/ISCI 1401 3.00 or SC/ISCI 1410 6.00.

Course credit exclusions: SC/MATH 1581, SC/MATH 2580 6.00, SC/MATH 2581 3.00, GL/ECON1950 3.0, GL/MATH 1950 3.0, GL/MATH 2680 6.00, GL/MODR 1950 3.0.

Note: This course is a prerequisite for SC/MATH 2281 3.00 and SC/MATH 3280 3.00.

A quantitative introduction to financial economics. The topics include arbitrage pricing theory, forwards and futures, American and European options, interest rate derivatives, yield curves, arbitrage hedging and pricing, put-call parity, arbitrage bounds, binomial model, Black-Scholes formula, risk-neutral valuation, trinomial model. The course ensures an adequate preparation for exam MFE of the Society of Actuaries.

Prerequisites: SC/MATH 2030 3.00; SC/MATH 2280 3.00

Note: This course is a prerequisite for SC/MATH 3282 3.00.

We begin the course with the calculus and geometry of vector-valued functions, i.e. curves, in 2-dimensional and 3-dimensional Euclidean spaces. Then we study differentiation and integration of functions defined on regions in the above-mentioned spaces. In differentiation, we study partial derivatives, tangent planes to surfaces in 3-dimensional Euclidean spaces, directional derivatives, the gradient, extrema of functions of 2 and 3 variables. In integration, we study double integrals in rectangular and polar coordinates, and triple integrals in rectangular, cylindrical and spherical coordinates. We end with changes of coordinates in double and triple integrals.

Prerequisite: SC/MATH 1014 3.00 or SC/MATH 1310 3.00 or SC/ISCI 1402 3.00 or SC/ISCI 1410 6.00. Students should have a knowledge of vector algebra in two and three dimensions.

Course credit exclusions: SC/MATH 2015 3.00, GL/MATH/MODR 2670 3.00, GL/MATH 3200 3.00.

Note: This course is a prerequisite for SC/MATH 2131 3.00, SC/MATH 2270 3.00, SC/MATH 2271 3.00, SC/MATH 3271 3.00, SC/MATH 3410 3.00, SC/MATH 4001 6.00, SC/MATH 4143 3.00, SC/MATH 4171 3.00, SC/MATH 4172 3.00 and SC/MATH 4271 3.00.

Statistics plays a key role in almost all areas of human inquiry. Its importance has grown considerably with the avail­ability of large amounts of data gathered elec­tronical­ly. This course presents an introduction to the concepts and methods of statistics including confidence intervals, tests of significance, re­gres­sion, analysis of variance and other methods. Students will learn how to use the statistical software R for data analysis.

Prerequisite: High school MATH 11U or MATH 11U/C.

Course credit exclusions: SC/MATH 2930 3.00, SC/BIOL 2060 3.00, AP/ECON 2500 3.00, AP/SC/GEOG 2420 3.00, HH/KINE 2050 3.00, SC/MATH 2560 3.00, SC/MATH 2570 3.00, HH/PSYC 2020 6.00, SB/OMIS 1000 3.00.

Note: Students who have passed SC/MATH 1131 3.00 may not take SC/MATH 2565 3.00.

This is an applied probability and statistics course for engineering students. The aim is to provide an application oriented introduction to probability and statistics. The examples will be from a wide selection of engineering disciplines. The probability component is about 30% of the lectures. About 40% of the time, the lectures and tutorials focus on solving practical statistical problems that emerge from engineering problems.

Prerequisites: SC/MATH 1014 3.00 or equivalent; SC/MATH 1025 3.00 or equivalent; LE/EECS 1011 3.00 or equivalent.

Course credit exclusions: SC/MATH 1131 3.00; SC/MATH 2560 3.00; SC/MATH 2570 3.00; SC/MATH 2565 3.00.