Course Descriptions - Mathematics
College of Engineering and Science
Mathematics and Statistics

4.14.2 COURSE DESCRIPTIONS-MATHEMATICS

All courses listed will not necessarily be offered each year.

62-100.Mathematical Foundations
Logic, sets, relations, functions. Development of skills in theoretical mathematics. (Prerequisite: 60-100 or 62-120.) (2 lecture, 2 tutorial hours a week.)

62-120.Linear Algebra I
Linear systems, matrix algebra, determinants, vectors in Rn, dot product, orthogonalization, eigenvalues, and diagonalization. (Prerequisite: OAC Algebra and Geometry or equivalent.) (Antirequisite: 62-126.) (3 lecture hours, 1 tutorial hour a week.)

62-126.Linear Algebra (Engineering)
Linear systems, matrix algebra, determinants, vectors in Rn, dot product, orthogonalization, and eigenvalues. (Prerequisite: OAC Algebra and Geometry, or equivalent.) (Antirequisite: 62-120.) (3 lectures hours, 1 tutorial hour a week.)

62-130.Elements of Calculus
Review of differentiation, exponential functions, and indefinite integrals. Methods of integration, differential equations, partial derivatives. A variety of applications. (Prerequisite: OAC Calculus.) (May not be taken for credit concurrently with, or subsequent to having obtained credit in 62-140. This course is not a sufficient prerequisite to 62-141, but may serve as preparation for 62-140.) (3 lecture hours, 1 tutorial hour a week.)

62-140.Calculus A
Limits and continuity; differential calculus with applications, extending OAC Calculus; related rates, differentials. Mean Value Theorem. Antiderivatives, Riemann sums and the definite integral. Fundamental Theorem of Calculus. Selected integration techniques. Selected applications of the definite integral. (Prerequisite: OAC Calculus, or equivalent.) (3 lecture hours, 1 tutorial hour a week.)

62-141.Calculus B
Indeterminate forms and l'Hopital's Rule. Further techniques of integration. Improper integrals, numerical integration. Separable differential equations. Further applications of definite integrals. Polar and parametric co-ordinates. Infinite sequences and series: tests for convergence, power series (Taylor, Maclaurin, binomial). (Prerequisite: 62-140) (3 lecture hours, 1 tutorial hour a week.)

62-194.Mathematics for Business
Mathematics of finance. Solutions of linear equations, matrices, linear inequalities, simplex method for linear programming. Probability theory. (This course is intended for students in the Faculty of Business Administration.) (Prerequisite: Any OAC Mathematics course.) (3 lecture hours, 1 tutorial hour a week.)

62-198.Ideas in Mathematics
Intended for students outside of Mathematics and Science. Selected topics from algebra, analysis, geometry, probability, and statistics. (Not available for credit for students in the College of Engineering and Science.) (3 lecture hours, 1 tutorial hour a week.)

62-210.Multivariable and Vector Differential Calculus
Review of vector functions of one variable. Differential calculus of functions of more than one variable. Vector differential calculus. Multiple integration. (Prerequisites: 62-141, and 62-120 or 62-126.) (Antirequisite: 62-215.) (3 lecture hours, 1 tutorial hour a week.)

62-211.Vector Integral Calculus and Differential Equations
Surface integrals, line integrals, and integral theorems. Ordinary differential equations and the Laplace transform. (Prerequisite: 62-210.) (Antirequisite: 62-216.) (3 lecture hours, 1 tutorial hour a week.)

62-212.Introduction to Analysis I
Real numbers. Limits, sequences, and continuity. Differentiation. (Prerequisites: 62-100, 62-120, and 62-141.) (3 lecture hours, 1 tutorial hour a week.)

62-213.Introduction to Analysis II
Sequences and series of functions. Uniform and absolute convergence. Power Series. Integration. (Prerequisite: 62-212.) (3 lecture hours, 1 tutorial hour a week.)

62-215.Vector Calculus
Quadric surfaces. Vector differential calculus. Multiple integration. Line and surface integrals. (Prerequisites: 62-141, and 62-120 or 62-126.) (Antirequisite: 62-210.) (3 lecture hours, 1 tutorial hour a week.)

62-216.Differential Equations
Differential equations and Laplace transforms. Series solution of differential equations. Applications to science and engineering. (Prerequisites: 62-141, and 62-120 or 62-126.) (Antirequisite: 62-211.) (3 lecture hours, 1 tutorial hour a week.)

62-218.Complex Variables
Complex numbers. Analytic functions. Contour integration. Series, Laurent expansions, residues. Application to real integrals. (Credit not allowed towards any major program in Mathematics.) (Prerequisite: 62-210 or 62-215; corequisite: 62-211 or 62-216.) (3 lecture hours, 1 tutorial hour per week.)

62-220.Linear Algebra II
Rigourous study of the following topics: linear systems, vector spaces, linear transformations, projections, pseudo-inverses, determinants, inner product spaces and applications. (Prerequisites: 62-100 and 62-120 .) (3 lecture hours, 1 tutorial hour a week.)

62-221.Linear Algebra III
A rigourous treatment of eigenvalues and eigenvectors, diagonalization, similarity problem and canonical form for real and complex matrices; positive definite matrices; computational methods for approximating solutions to systems of linear equations and eigenvalues. (Prerequisite: 62-220.) (3 lecture hours, 1 tutorial hour a week.)

62-222.Number Theory
Divisibility, congruences, numerical functions. Theorems of Euler, Fermat, and Wilson. Theory of primes and quadratic residues. (Prerequisites: 62-100 and 62-120.) (3 lecture hours a week.)

62-240.Combinatorics
Finite combinatorics; counting problems involving set operations, relations and functions; principle of inclusion and exclusion; ordinary and exponential generating functions; recurrence relations. (Prerequisites: 62-100 and 62-141.) (3 lecture hours a week.)

62-292.Theory of Interest
Measurement of interest, elementary and general annuities, amortization schedules and sinking funds, bonds, depreciation, depletion, and capitalized cost. This course helps prepare students for the Society of Actuaries examinations. (Prerequisite: 62-141 or consent of instructor.) (3 lecture hours a week.)

62-310.Principles of Analysis I
Metric spaces. Countability, compactness, and connectedness. Differentiation of functions of several variables. Implicit and inverse function theorems. (Prerequisite: 62-213.) (3 lecture hours a week.)

62-312.Complex Analysis
Analytic functions. Power series. Elementary functions. Contour integration. Cauchy theorem. Singularities. Residues. Laurent expansions. (Prerequisites: 62-212 and one of 62-211 or 62-215.) (3 lecture hours a week.)

62-313.Applied Complex Analysis
Applications of residue theorem, conformal mapping, and analytic continuation. Poisson kernel, harmonic functions, and asymptotic expansions. (Prerequisite: 62-312 or consent of instructor.) (3 lecture hours a week.)

62-321.Abstract Algebra
Introduction to groups, rings, and fields. (Prerequisite: 62-220 or 62-222.) (3 lecture hours a week.)

62-324.Applied Algebra
Coding theory in cryptography and informatics; combinatorial designs and finite geometrics. (Prerequisite: 62-222; 62-321 is recommended.) (3 lecture hours a week.)

62-330.Computational Geometry
Homogeneous co-ordinates and point transformations. Curves, surfaces, and regions. Intersection, curve/surface fitting, and surface patches. Application to computer graphics, computer-aided geometric/cartographic design, pattern recognition, and image processing. (Prerequisites: 62-120 and 62-210 or 62-215.)

62-332.Tensor Analysis
Tensor algebra. Covariant differentiation. Tensor form of gradient, divergence, and curl. Riemann-Christoffel symbols. Curvature tensor. Applications. (Prerequisites: 62-210 and 62-211, or 62-215 and 62-216.) (3 lecture hours a week.)

62-338.Differential Geometry
Regular curves. Curvature, torsion, and Frenet equations. Surfaces, tangent planes, and normal vectors. Distance element. Classification of elliptic, hyperbolic, parabolic, and Euclidean surfaces. (Prerequisite: 62-210 or 62-215.) (3 lecture hours a week.)

62-360.Special Functions
Uniform convergence, Fourier Series, Orthonormal bases, Sturm-Liouville eigenvalue problems, eigenfunction expansions, Gamma function, Bessel functions, Legendre polynomials and functions, and the hypergeometric functions. (Prerequisite: 62-211, or 62-215 and 62-216.) (3 lecture hours a week.)

62-361.Dynamical Systems
An introduction to simple dynamical systems, both discrete and continuous. Long-term behaviour of such systems. Stability, periodicity, and chaos. A brief treatment of fractals. (Prerequisite: 62-120, 62-211, or 62-216.) (3 lecture hours a week.)

62-371.Continuum Mechanics
Cartesian tensors and analysis. Continuum. Kinematics. Equations of motion. Applications to fluids and solids. (Prerequisites: 62-210 and 62-211, or 62-215 and 62-216.) (3 lecture hours a week.)

62-374.Linear Programming
Topics covered are: geometric linear programming, the Simplex method, the revised Simplex method, duality theory, sensitivity analysis, project planning and integer programming. Optional topics include: the transportation problem, the upper bounding technique, the dual Simplex method, parametric linear programming, game theory, and goal planning. Completion of some assignments will require the use of computer software packages. This course is intended to help students prepare for some parts of the Society of Actuaries examination on Operations Research (Course 130). Interested students should also take 65-376. (Prerequisite: 62-220 or consent of instructor.) (Antirequisite: 91-312.) (3 lecture hours a week.)

62-380.Numerical Methods
Topics covered are: nonlinear equations in one variable, interpolation, numerical integration (quadrature), and linear systems (direct methods). Optional topics are: numerical differentiation, iterative methods for boundary value problems. Completion of some assignments will require the use of computer software packages. This course is intended to help students prepare for some parts of the Society of Actuaries examination on Numerical Methods (Course 135). (Prerequisites: 62-210 or 62-215, 62-211 or 62-216. 62-120 or 62-126.) (May not be taken for credit after 62-481.) (3 lecture hours a week.)

62-400.Mathematical Logic
Propositional logic. Proof theory and model theory of first-order logic. Undecidability. (Prerequisite: 62-213 or 62-221.) (3 lecture hours a week.)

62-401.Axiomatic Set Theory
Zermelo-Fraenkel axioms. Ordinal and cardinal numbers. Founding mathematics on set theory. Selected topics. (Prerequisite: 62-213 or 62-221.) (3 lecture hours a week.)

62-410.Real Analysis I
Lebesgue measure and Lebesgue integral. Differentiation and integration. Radon-Nikodym theorem. (Prerequisite: 62-213.) (3 lecture hours a week.)

62-411.Real Analysis II
Metric spaces. Topological spaces. Stone-Weierstrass and Ascoli theorems. Classical Banach spaces. (Prerequisite: 62-410.) (3 lecture hours a week.)

62-420.Introduction to Group Theory
Abstract groups, subgroups, isomorphism theorems, orbits, class equation, quotient groups, Sylow's theorems, metric vector spaces, quadratic forms, basic concepts of orthogonal geometry, the classical groups. (Prerequisites: 62-221 and 62-321.) (3 lecture hours a week.)

62-421.Introduction to Ring Theory
Matrix rings, polynomial rings, fields of fractions, principal ideal domains and Euclidean domains, finitely generated modules over a p.i.d. (Prerequisites: 62-221 and 62-321.) (3 lecture hours a week.)

62-422.Introduction to Field Theory
Polynomial rings, splitting fields, The Fundamental Theorem of Galois Theory, Galois' criterion for solvability by radicals, algebraically closed fields, finite fields. (Prerequisites: 62-221 and 62-321.) (3 lecture hours a week.)

62-434.Point Set Topology
Topological spaces. Neighbourhood systems. Homomorphisms. Product and quotient spaces. Separation axioms. Compactness, connectedness. Metric spaces: convergence, completeness, category. (Prerequisite: 62-213.) (3 lecture hours a week.)

62-460.Applied Mathematics Methods I
General basic concepts for linear partial differential equations. Classification of second-order equations and canonical forms. An introduction to theory of distribution. Sturm-Liouville theory for ODEs. Fourier series and integral transforms with applications to PDEs. (Prerequisites: 62-218 or 62-312, and 62-360.) (3 lecture hours a week.)

62-461.Applied Mathematics Methods II
Qualitative and quantitative analysis of hyperbolic, parabolic, and elliptic partial differential equations. (Prerequisite: 62-460.) (3 lecture hours a week.)

62-470.Fluid Dynamics I
Kinematics, stress hypothesis, constitutive equations, equations of motion. Ideal fluid flow in two and three dimensions. Introduction to potential theory and use of complex variable theory. Effects of viscosity and compressibility. Introduction to computational problems in two-dimensions. (Prerequisites: 62-210 and 62-211, or 62-215 and 62-216, and 62-218 or 62-312.)

62-471.Fluid Dynamics II
Navier-Stokes equations for viscous incompressible flows, exact solutions, boundary layer theory, and asymptotic methods. Compressible inviscid flows, one-dimensional unsteady flows, two-dimensional irrotational flows, method of characteristics. Introduction to shock waves. (Prerequisite: 62-470.) (3 lecture hours a week.)

62-472.Solid Mechanics
Theory of mechanics of solid continuum, including elasticity, plasticity, and viscoelasticity. (Prerequisites: 62-210 and 62-211 or 62-215 and 62-216, and 62-218 or 62-312.) (3 lecture hours a week.)

62-480.Numerical Linear Algebra
Topics include: floating point arithmetic, matrix factorizations, condition number of matrices, iterative methods, eigenproblems, singular value decomposition. Completion of some assignments will require computer programming and/or the use of major software packages. (Prerequisites: 62-221 and 60-141.) (3 lecture hours a week.)

62-481.Numerical Analysis
Topics include: floating point arithmetic, solution of nonlinear algebraic equations, polynomial and spline interpolation, functional approximation, numerical differentiation and integration, numerical solution of ordinary differential equations, unconstrained minimization. Completion of some assignments will require computer programming and/or the use of major software packages. (Prerequisites: 62-211 and 62-480.) (3 lecture hours a week.)

62-482.Mathematical Programming
Topics include: unconstrained optimization, convexity, least squares problems, optimality conditions, penalty methods. Completion of some assignments will require the use of computer software packages. (Prerequisites: 62-210, 62-212, 62-221, and one of 62-374, 62-380, or 65-376.) (3 lecture hours a week.)

62-490.Actuarial Mathematics I
Life contingencies. Survival distributions and life tables, life insurance, life annuities, net premiums, net premium reserves. This course helps prepare students for the Society of Actuaries examinations. (Prerequisites: 62-292, 65-251, 62-215, and 62-216 or consent of instructor.) (3 lecture hours a week.)

62-492.Actuarial Mathematics II
Selection of topics from: advanced life contingencies, risk theory, survival models, construction and graduation of mortality tables. This course helps prepare students for the Society of Actuaries examinations. (Prerequisite: 62-490 or consent of instructor.) (3 lecture hours a week.)

62-498.Topics in Mathematics
Advanced topics not covered in other courses. (May be repeated for credit when the topic is different.) (Prerequisite: consent of the instructor.) (3 lecture hours a week.)