A review of these particular topics for students who have completed Mathematics 211 or equivalent. Course Hours:E(8 hours) Notes:Open to students with credit in Mathematics 211 or equivalent. NOT INCLUDED IN GPA

Multiple Integration and applications. Course Hours:E(16 hours) Prerequisite(s):Mathematics 253 or 283 or Applied Mathematics 219; or consent of the Department. Notes:Designed to rectify a deficiency for those students whose Calculus I and II courses did not cover the multivariate topics from Applied Mathematics 219. NOT INCLUDED IN GPA

Inverse functions and inverse trigonometric functions. Hyperbolic and inverse hyperbolic functions. Indeterminate forms. Applications of integration. Course Hours:E(8 hours) Prerequisite(s):Mathematics 249 or 251 or 281 or Applied Mathematics 217; or consent of the Department. Notes:Designed to rectify a deficiency for those students whose first Calculus course did not cover some of the topics from Applied Mathematics 217. NOT INCLUDED IN GPA

Vector functions and differentiation, curves and parametrization, functions of several variables, partial differentiation, differentiability, implicit functions, extreme values. Course Hours:E(16 hours) Prerequisite(s):Mathematics 267 or consent of the Department. Notes:Designed to rectify a deficiency for those students whose Calculus I and II courses covered all the topics from Mathematics 265 and 267 but did not cover some of the topics on the calculus of functions of several variables from Mathematics 277. NOT INCLUDED IN GPA

Note: Students who have not studied mathematics for some time are strongly advised to review high school material thoroughly prior to registering in any junior level mathematics course.

Mathematics 205

Mathematical Explorations

A mathematics appreciation course. Topics selected by the instructor to provide a contemporary mathematical perspective and experiences in mathematical thinking. May include historical material on the development of classical mathematical ideas as well as the evolution of recent mathematics. Course Hours:H(3-1) Prerequisite(s):Mathematics 30-1, Mathematics 30-2, Pure Mathematics 30, Applied Mathematics 30, or Mathematics II (offered by Continuing Education). Notes:For students whose major interests lie outside the sciences. Highly recommended for students pursuing an Elementary School Education degree. It is not a prerequisite for any other course offered by the Department of Mathematics and Statistics, and cannot be used for credit towards any Major or Minor program in the Faculty of Science except for a major in General Mathematics.

Systems of equations and matrices, vectors, matrix representations and determinants. Complex numbers, polar form, eigenvalues, eigenvectors. Applications. Course Hours:H(3-1T-1) Prerequisite(s):A grade of 70 per cent or higher in Mathematics 30-1 or Pure Mathematics 30. (Alternatives are presented in C.1 Mathematics Diagnostic Test in the Academic Regulations section of this Calendar). Antirequisite(s):Credit for Mathematics 211 and either 213 or 221 will not be allowed.

Systems of equations and matrices, vectors, linear transformations, determinants, eigenvalues and eigenvectors. Course Hours:H(3-1T-1) Prerequisite(s):A grade of 70 per cent or higher in Mathematics 30-1 or Pure Mathematics 30. Antirequisite(s):Credit for Mathematics 213 and 211 will not be allowed.

Algebraic operations. Functions and graphs. Limits, derivatives, and integrals of exponential, logarithmic and trigonometric functions. Fundamental theorem of calculus. Applications. Course Hours:H(4-1T-1) Prerequisite(s):A grade of 70 per cent or higher in Mathematics 30-1 or Pure Mathematics 30. (Alternatives are presented in C.1 Mathematics Diagnostic Test in the Academic Regulations section of this Calendar). Antirequisite(s):Not open to students with 60 per cent or higher in Mathematics 31, except with special departmental permission. Credit for more than one of Mathematics 249, 251, 265, 275, 281, or Applied Mathematics 217 will not be allowed.

Inverses of trigonometric functions. Methods of integration, improper integrals. Separable differential equations, first and second order linear differential equations, applications. Course Hours:H(3-1T-1) Prerequisite(s):Mathematics 249 or 251 or 281 or Applied Mathematics 217. Antirequisite(s):Credit for more than one of Mathematics 253, 283, or Applied Mathematics 219 will not be allowed. Notes:Mathematics 253 or 283 is a prerequisite for many 300-level courses in Pure Mathematics, Applied Mathematics, Statistics and Actuarial Science. Students in programs offered by the Department of Mathematics and Statistics are strongly recommended to take Mathematics 283. Mathematics 253 will be offered the last time in 2014.

Limits, derivatives, and integrals; the calculus of exponential, logarithmic, trigonometric and inverse trigonometric functions. Applications including curve sketching, optimization, exponential growth and decay, Taylor polynomials. Fundamental theorem of calculus. Improper integrals. Introduction to partial differentiation. Course Hours:H(3-1T-1) Prerequisite(s):A grade of 70 per cent or higher in Mathematics 30-1 or Pure Mathematics 30; and a grade of 50 per cent or higher in Mathematics 31. (Alternatives to Pure Mathematics 30 are presented in C.1 Mathematics Diagnostic Test in the Academic Regulations section of this Calendar). Antirequisite(s):Credit for more than one of Mathematics 249, 251, 265, 275, 281, or Applied Mathematics 217 will not be allowed. Notes:This course provides the basic techniques of differential calculus as motivated by various applications. Students performing sufficiently well in a placement test may be advised to transfer directly to Mathematics 267.

Sequences and series, techniques of integration, multiple integration, applications; parametric equations. Course Hours:H(3-1T-1) Prerequisite(s):Mathematics 249 or 251 or 265 or 275 or 281 or Applied Mathematics 217. Antirequisite(s):Credit for more than one of Mathematics 267, 277, 349, or Applied Mathematics 219 will not be allowed.

Proof techniques. Sets and relations. Induction. Counting and probability. Graphs and trees. Course Hours:H(3-1T-1) Prerequisite(s):Mathematics 30-1 or Pure Mathematics 30. Antirequisite(s):Credit for both Mathematics 271 and 273 will not be allowed. Notes:Philosophy 279 or 377 is highly recommended to complement this course.

Introduction to proofs. Functions, sets and relations. The integers: Euclidean division algorithm and prime factorization; induction and recursion; integers mod n. Real numbers: sequences of real numbers; completeness of the real numbers; open and closed sets. Complex numbers. Course Hours:H(3-1T-1) Prerequisite(s):A grade of 80 per cent or higher in Mathematics 30-1 or Pure Mathematics 30. (Alternatives are presented in C.1 Mathematics Diagnostic Test in the Academic Regulations section of this Calendar). Antirequisite(s):Credit for both Mathematics 273 and 271 will not be allowed.

Calculus of functions of one real variable; derivative and Riemann integral; Mean Value Theorem; the Fundamental Theorem of Calculus; techniques of integration; Applications; Improper integrals; Power series, Taylor series. Course Hours:H(3-1T-1.5) Prerequisite(s):A grade of 70 per cent or higher in Pure Mathematics 30 or Mathematics 30-1; and credit in Mathematics 31. Alternatively, admission to the Faculty of Engineering including credit in either Pure Mathematics 30 or Mathematics 30-1; and Mathematics 31. Antirequisite(s):Credit for more than one of Mathematics 249 or 251 or 265 or 275 or 281 or Applied Mathematics 217 will not be allowed.

Multivariable Calculus for Engineers and Scientists

Calculus of functions of several real variables; differentiation, implicit functions, double and triple integrals; applications; Vector-valued functions; derivatives and integrals; parametric curves. Course Hours:H(3-1T-1.5) Prerequisite(s):Mathematics 275 or Applied Mathematics 217; or consent of the Department. Antirequisite(s):Credit for more than one of Mathematics 253 or 267 or 277 or 283 or Applied Mathematics 219 will not be allowed.

Vector spaces and subspaces. Linear independence. Matrix representations of linear transformations. Gram-Schmidt orthogonalization. Students will complete a project using a computer algebra system. Course Hours:H(3-1T) Prerequisite(s):One of Mathematics 211 or 213. Antirequisite(s):Credit for both Mathematics 311 and 313 will not be allowed.

Diagonalization. Canonical forms. Inner products, orthogonalization. Spectral theory. Students will be required to complete a project using a computer algebra system. Course Hours:H(3-1T) Prerequisite(s):Mathematics 213 or a grade of "B+" or better in Mathematics 211. Antirequisite(s):Credit for both Mathematics 311 and 313 will not be allowed.

Systems of ordinary differential equations. Calculus of functions of several variables. Introduction to vector analysis, theorems of Green, Gauss and Stokes. Course Hours:H(3-1T) Prerequisite(s):One of Mathematics 253 or 267 or 277 or 283 or Applied Mathematics 219; and Mathematics 211 or 213. Antirequisite(s):Credit for more than one of Mathematics 331 or 353 or 367 or 377 or 381 or Applied Mathematics 309 will not be allowed. Notes:This course is not a member of the list of courses constituting the fields of Actuarial Science, Applied Mathematics, Pure Mathematics, or Statistics and cannot normally be substituted for Mathematics 353 or 367 or 377 or 381 in degree programs in any of those fields.

The real numbers, sequences, series, functions, continuity and uniform continuity, differentiation, intermediate and mean value theorems, the Riemann integral, integrability of continuous functions on closed intervals. Course Hours:H(3-1T) Prerequisite(s):Mathematics 253 or 267 or 277 or 283 or Applied Mathematics 219; or consent of the Department. Antirequisite(s):Credit for more than one of Mathematics 335, 355, Pure Mathematics 435 or 455 will not be allowed. Notes:Students with a grade of "B+" or higher in Mathematics 267 or 277 are encouraged to consider taking Mathematics 355.

Infinite sequences and series. Polar co-ordinates, parametric equations, arc length. Vector geometry, differentiation of vector-valued functions. Partial differentiation. Students will complete a project using a computer algebra system. Course Hours:H(3-1T) Prerequisite(s):Mathematics 253 or 283 or Applied Mathematics 219; and Mathematics 211 or 213. Antirequisite(s):Credit for both Mathematics 349 and 381 will not be allowed. Notes:This course will not be offered after Winter 2015.

Applications of partial differentiation, multiple integrals, vector calculus including Stokes' and the Divergence Theorems. Course Hours:H(3-1T) Prerequisite(s):Mathematics 349. Antirequisite(s):Credit for more than one of Mathematics 353, 331, 381 or Applied Mathematics 309 will not be allowed. Notes:This course will not be offered after Winter 2015.

The real numbers, sequences, series, functions, continuity and uniform continuity, differentiation, intermediate and mean value theorems, the Riemann integral, integrability of continuous functions on closed intervals. Course Hours:H(3-1T) Prerequisite(s):Mathematics 283 or 267 or 277; or a grade of "B+" or better in Mathematics 253 or Applied Mathematics 219. Antirequisite(s):Credit for more than one of Mathematics 335, 355, Pure Mathematics 435 and 455 will not be allowed.

Functions of several variables; limits, continuity, differentiability, partial differentiation, applications including optimization and Lagrange multipliers. Vector functions, line integrals and surface integrals, Green’s theorem, Stokes’ theorem. Divergence theorem. Students will complete a project using a computer algebra system. Course Hours:H(3-1T) Prerequisite(s):One of Mathematics 267 or 283 or 349 or Applied Mathematics 219; and Mathematics 211 or 213. Antirequisite(s):Credit for more than one of Mathematics 353, 331, 367, 377, 381 or Applied Mathematics 309 will not be allowed.

Differential Equations for Engineers and Scientists

Definition, existence and uniqueness of solutions; first order and higher order equations and applications; Homogeneous systems; Laplace transform; partial differential equations of mathematical physics. Course Hours:H(3-1.5T) Prerequisite(s):Applied Mathematics 219 or Mathematics 277; or both Mathematics 267 and 177; or both Mathematics 253 and 114. Antirequisite(s):Credit for more than one of Mathematics 375 or Applied Mathematics 307 or 311 will not be allowed.

Review of calculus of functions of several variables. Vector fields, line integrals, independence of path, Green’s theorem; Surface integrals, divergence theorem, Stokes’s theorem; applications; curvilinear coordinates; Laplace, diffusion and wave equations in three dimensional space. Course Hours:H(3-1.5T) Prerequisite(s):Mathematics 375. Antirequisite(s):Credit for more than one of Mathematics 377, 331, 353, 367, 381 or Applied Mathematics 309 will not be allowed.

Functions of several variables; differentiability, extrema. Implicit and inverse function theorems. Integration of functions of several variables; line integrals; surface integrals. Students will complete a project using a computer algebra system. Course Hours:H(3-1T) Prerequisite(s):Mathematics 283 or a grade of "B+" or better in Mathematics 253 or Applied Mathematics 219; and Mathematics 211 or 213. Antirequisite(s):Credit for Mathematics 381 and any one of Mathematics 331, 349, 353, and Applied Mathematics 309 will not be allowed. Notes:This course will not be offered after Winter 2015.

Higher level topics which can be repeated for credit. Course Hours:H(3-0) Prerequisite(s):Consent of the Department. Notes:This course is designed to add flexibility to completion of an undergraduate pure mathematics or general mathematics program. MAY BE REPEATED FOR CREDIT

Techniques of integration. Multiple integrals. Analysis of functions. Continuity. Compact sets. Convex sets. Separating hyperplanes. Lower and upper hemi-continuous correspondences. Fixed point theorems, Optimal control. Course Hours:H(3-0) Prerequisite(s):Mathematics 211 or 213; and Mathematics 253 or 267 or 277 or 283 or Applied Mathematics 219. Alternatively, both Economics 387 and 389.

Canonical forms. Inner product spaces, invariant subspaces and spectral theory. Quadratic forms. Course Hours:H(3-1T) Prerequisite(s):Mathematics 311; and one of Mathematics 331, 353, 367, 377, 381 or Applied Mathematics 309. Antirequisite(s):Credit for more than one of Mathematics 411, 313 or Applied Mathematics 441 will not be allowed. Notes:May not be offered every year. Consult the Department for listings.

Basic complex analysis – complex numbers and functions, differentiation, Cauchy-Riemann equations, line integration, Cauchy’s theorem and Cauchy’s integral formula, Taylor’s theorem, the residue theorem, applications to computation of definite integrals. Course Hours:H(3-1T) Prerequisite(s):Both Mathematics 349 and 353; or both Mathematics 283 and 381; or Mathematics 267. Antirequisite(s):Credit for more than one of Mathematics 421, 423, Pure Mathematics 421 or 521 will not be allowed.
Notes:Students with credit in Mathematics 267 are strongly recommended to take Mathematics 367 before or while taking Mathematics 421.

Basic complex analysis – complex numbers and functions, differentiation, Cauchy-Riemann equations, line integration, Cauchy’s theorem and Cauchy’s integral formula, Taylor’s theorem, the residue theorem, applications to computation of definite integrals. Course Hours:H(3-1T) Prerequisite(s):Both Mathematics 349 and 353; or both Mathematics 283 and 381; or Mathematics 267. Antirequisite(s):Credit for more than one of Mathematics 421, 423, Pure Mathematics 421 or 521 will not be allowed. Notes:Open only to Honours Applied Mathematics and Honours Pure Mathematics students. Students with credit in Mathematics 267 are strongly recommended to take Mathematics 367 before or while taking Mathematics 423.

Basic topology of Euclidean space, Fubini’s theorem, the total derivative, change of variable in multiple integrals, inverse and implicit function theorems, submanifolds of Euclidean spaces, differential forms, Stokes’ theorem in arbitrary dimension. Course Hours:H(3-0) Prerequisite(s):Mathematics 353 or 367 or 377 or 381 or Applied Mathematics 309; and Mathematics 311 or 313; and Mathematics 335 or 355 or Pure Mathematics 435 or 455; or consent of the Department. Antirequisite(s):Credit for more than one of Mathematics 445, 447 or Pure Mathematics 545 will not be allowed.

Basic topology of Euclidean space, Fubini’s theorem, the total derivative, change of variable in multiple integrals, inverse and implicit function theorems, submanifolds of Euclidean spaces, differential forms, Stokes’ theorem in arbitrary dimension. Course Hours:H(3-0) Prerequisite(s):Mathematics 367 or 377 or 381 or Applied Mathematics 309 or "B+" or higher in Mathematics 353; and Mathematics 313 or "B+" or higher in Mathematics 311; and Mathematics 355 or Pure Mathematics 455 or "B+" or higher in Mathematics 335 or Pure Mathematics 435. Alternatively, consent of the Department. Antirequisite(s):Credit for more than one of Mathematics 445, 447 or Pure Mathematics 545 will not be allowed.

Abstract measure theory, basic integration theorems, Fubini's theorem, Radon-Nikodym theorem, Lp Spaces, Riesz representation theorems. Course Hours:H(3-0) Prerequisite(s):Mathematics 545 or Pure Mathematics 545 or consent of the Department. Antirequisite(s):Credit for more than one of Mathematics 501, 601, Pure Mathematics 501 or 601 will not be allowed.

Analytic functions as mappings, local properties of analytic functions, Schwarz lemma, Casorati-Weierstrass and Picard theorems, analytic continuation, harmonic and subharmonic functions, approximation theorems, conformal mappings, Riemann surfaces. Course Hours:H(3-0) Prerequisite(s):Mathematics 335 or 355 or Pure Mathematics 435 or 455; and Mathematics 421 or 423 or Pure Mathematics 421; or consent of the Department. Antirequisite(s):Not open to students with credit in Pure Mathematics 521.

Sequences and series of functions; Lebesgue integration on the line, Fourier series and the Fourier transform, pointwise convergence theorems, distributions and generalized functions. Course Hours:H(3-0) Prerequisite(s):Mathematics 447 or a grade of "B+" or better in Pure Mathematics 445 or Mathematics 445. Antirequisite(s):Not open to students with credit in Pure Mathematics 545.

Graduate CoursesNote: In addition to the prerequisites listed below, consent of the Applied Mathematics Department or the Pure Mathematics Department is a prerequisite for these graduate courses.

Mathematics 601

Measure and Integration

Abstract measure theory, basic integration theorems, Fubini's theorem, Radon-Nikodym theorem, Lp spaces, Riesz representation theorem. Course Hours:H(3-0) Prerequisite(s):Mathematics 545 or Pure Mathematics 545 or consent of the Department. Antirequisite(s):Credit for more than one of Mathematics 501, 601, Pure Mathematics 501 or 601 will not be allowed.

Analytic functions as mappings, local properties of analytic functions, Schwarz lemma, Casorati-Weierstrass and Picard theorems, analytic continuation, harmonic and subharmonic functions, approximation theorems, conformal mappings, Riemann surfaces. Course Hours:H(3-0) Prerequisite(s):Mathematics 335 or 355 or Pure Mathematics 435 or 455; or consent of the Department.