Applied Mathematics (AM)

AM 380B AM 481C

AM 381 AM 351 AM 389 deleted AM 391 AM 451 AM 395 AM 3710 AM 399 deleted AM 444 AM 435 AM 462 AM 431 AM 463 deleted AM 464 deleted AM 465A AM 373 AM 465B AM 473 AM 466 AM 4630 AM 468 AM 491 AM 472 AM 432 AM 474 AM 433

AM 476 AM 464C

AM 478 AM 492 AM 481A/B AM 453 AM 482 AM 456 AM 485 deleted AM 486 AM 477 AM 488 AM 455 AM 489 AM 495

AM 495 AM 465C

AM 499 AM 496

AM 430 AM 493/4 (not offered)

Listed without a description.

AM 231 Calculus 4F,W 3C,1T 0.5

Vector integral calculus, including line integrals, Green's theorem, the Divergence theorem, and Stokes’ theorem, with applications to physical problems. Introduction to the Calculus of Variations. Sequences and series of functions and their applications, including the role of uniform convergence.

Prereq: MATH 237

Antireq: MATH 210, 212, 213B, 2208, 230B, 240B

Formerly MATH 2308

This courses replaces MATH 230B effective Winter, 1991.

AM 250 Modelling with Ordinary Differential EquationsF,W 3C 0.5

Overview of the modelling process. Examples of physical systems leading to ordinary differential equations. Applications to Newton's laws of motion, mechanical vibrations, and population dynamics. The emphasis is on the physical derivation and interpretation of the model equations.

Prereq: MATH 130B or 137

Antireq: MATH 215, 216

Formerly AM 260

AM 251 Modelling with Systems of Ordinary Differential EquationsF,W 3C 0.5

Simplification procedures in modelling. Physical dimensions, scaling, and the principle of dimensional homogeneity. Buckingham's Pi Theorem. Systems of ordinary differential equations. Phase plane analysis. Stability, The emphasis is on two-dimensional physical systems based on the mechanics of particles and ecological systems. Prereq: MATH 138 or 230A, or consent of instructor Formerly AM 270

AM 252 Foundations of Applied MathematicsF 3C 0.5

An in-depth look at some of the fundamental principles of applied mathematics, through case studies chosen from classical mechanics of discrete and continuum systems and diffusion processes. Group-theoretic foundations of dimensional analysis. Regular and singular perturbation theory.

Prereq: AM 251 and one of MATH 240A, 247 or AM 251 and one of MATH 230A, 237 plus consent of instructor

Formerly AM 280

Course Descriptions - 16:99 Mathematics.

AM 331 Real AnalysisF,W 3C 0.5

Topology of R®, continuity, norms, metrics, completeness. Fourier series and applications, for example, to ordinary differential equations, the heat problem, optimal approximation, the isoperimetric inequality. . Prerag: MATH 230B or 237 Antireq: MATH 322A, PMATH 351 Cross-listed as PMATH 331 Not available for credit to students in Honours Pure Mathematics programs.

AM 332 Complex AnalysisW,S 3C 0.5

Complex numbers; continuity, differentiability, analyticity of functions; the Cauchy-Riemann equations: solution of Laplace's equation; conformal mapping by elementary functions, and applications; contour tntegration, the Cauchy and allied theorems; Taylor and Laurent expansions, uniform convergence and power series; the residue calculus, and applications.

Prereq: MATH 2308 or 237

Antireq: MATH 322B, 332B, PMATH 352

Cross-listed as PMATH 332

Formerly MATH 332B

Not available for credit to students in Honours Pure Mathematics programs.

AM 333 Elementary Differential Geometry and Tensor Analysis Curves in Euclidean 3-Space (E%) andF,S 3C 0.5

the Serret-Frenet formulae; surfaces in E® and their intrinsic geometry. Gaussian curvature and the Gauss-Bonnet theorem. Co-ordinate transformations and tensors in n-dimensions; n-dimensional Riemannian spaces; covariant differentiation; geodesics; the curvature, Ricci and Einstein tensors. Applications of tensors in Relativity and Continuum Mechanics.

Prereq: AM 231 or MATH 2308, or consent of instructor

Cross-listed as PMATH 365

Formerly AM 362

AM 343 Discrete Models in Applied Mathematics Difference equations, Lapiace and zW 3C 0.5

transforms applied to discrete (and continuous) mathematical models taken from ecology, biology, economics and other fields.

Prereq: MATH 220B or 227, or consent of instructor

Formerly AM 340

16:100 Course Descriptions Mathematics: Applied Mathematics

AM 351 Ordinary Differential EquationsF,S 3C 0.5

Existence and uniqueness theorems; higher order and systems of equations; series solutions and special functions. Laplace transforms. Eigenvalues and eigenfunction expansions; applications to mathematical physics. Sturm's comparison, separation and oscillation theorems.

Prereq: MATH 230B or 237. AM 250 is recommended for non-AM majors.

Formerly AM 381

AM 353 Partial Differential Equations 1F,W,S 3C 0.5

First order partial differential equations and characteristic curves. Second order linear partial differential equations, primarily in two variables: physical origins; classification into hyperbolic, parabolic and ellidtic equations; the Cauchy initial value problem and characteristic curves. Derivation and analysis of solutions of the wave equation, heat equation and Laplace's equation, separation of variables and eigenfunction expansions; Fourier integrals; d'Alembert's solution and the propagation of waves; maximum principle for harmonic functions. Introduction to systems of partial differential equations.

hyperbolic systems, reduction to canon-

ical form.

Prereq: AM 231 and AM 251, or AM 251 and MATH 2308, or consent of instructor

Coreq: AM 351

Formerly AM 371

AM 361 Continuum MechanicsW 3C 0.5

Stress and strain tensors; analysis of stress and strain. Lagrangian and Eulerian methods for describing flow. Equations of continuity, motion and energy, constitutive equations. Navier-Stokes equation. Basic equations of etasticity. Various applications.

Prereq: AM 231 or MATH 2308, or consent of instructor

Coreq: AM 353 and AM/PMATH 332 (or PMATH 352)

Formerly AM 365

AM 371C Classical MechanicsF,S 3C 0.5

Kinematics and dynamics of systems of particles. Linear and angular momenta. Moving frames of reference. Moments of inertia. Kinematics and dynamics of rigid body motion. Virtual work. Lagrange's equations of motion. Hamilton's principle. Canonical equations. Conservation laws. Lagrange and Poisson bracket relations. Applications to problems in mechanics are stressed throughout.

Prereq: AM 231 or MATH 2308, or consent of instructor

Formerly AM 395

AM 373 Quantum Mechanics 1W 3C 0.5

Critical experiments and old quantum theory. Basic concepts of quantum mechanics: Observables, wavefunctions, Hamiltonians and the Schrédinger equation. Uncertainty, correspondence and super-position principles. Simple applications to finite and extended one-dimensional systems, harmonic oscillators, rigid rotor and hydrogen atom.

Prereq: AM 371C or consent of instructor

Formerly AM 465A

AM 375 Special Relativity and Electromagnetic Field Theory Minkowski space-time and LorentzW 3C 0.5

transformations. Physical consequences. Optics including Doppler effect and observation. Relativistic mechanics. Collisions. E = mc?. introduction to electricity and magnetism. Maxwell's equations. Tensorial formulation. Four-vector potential and gauge invariance. Algebraic structure of the electromagnetic field. Solutions of Maxwell's equations. Radiation: wave guides and antennae. Energy momentum tensor of the electromagnetic field.

Prereq: AM 333 and AM 371C, or consent of instructor

Formerly AM 364

AM 381C Introduction to Information TheoryF,S 3C 0.5

Variable length coding. The Shannon entropy as a measure of uncertainty and expected information. Minimal average length coding; the Shannon entropy as lower bound. Source entropy. Channels. Transinformation. Capacity. Applications to problem solving, information transmission, logics, science, linguistics and communications (TV, music, etc.). Determination of practical measures of information.

Prereq: Consent of instructor

Cross-listed as PMATH 380A

Formerly AM 380A

AM 431 F 3C 0.5

Measure and integration Lebesgue measure and integral for the real line, general measure and integration theory, convergence theorems, Fubini's theorem, absolute continuity, Radon Nikodym theorem, L?-spaces. Prereq: AM/PMATH 331 or PMATH 351 Cross-listed as PMATH 451 Formerly AM 462

AM 432 Functionat AnalysisW 3C 0.5

Banach spaces, linear operators, geometry of Hilbert spaces, Hahn- Banach theorem, open mapping theorem, compact operators, applications. Prereq: AM 431/PMATH 451 or PMATH 353 Cross-listed as PMATH 453 Formerly AM 472

AM 433 Ditferential GeometryF,S 3C 0.5

Some global aspects of surface theory, the Evler-Poincaré characteristic, the global interprdtation of Gaussian curvature via the Gauss-Bonnet formula. Submanifolds of E", induced Riemannian metrics, extrinsic and intrinsic curvatures, Gauss-Codazzi equations. Local Lie groups of transformations on R®, infinitesimal generators, the Lie derivative. An introduction to differentiable manifolds, the tangent and cotangent bundles, affine connections and the Riemann curvature tensor. The above topics will be illustrated by applications to continuum mechanics and mathematical physics.

Prereq: AM 333/PMATH 365 or consent of instructor

Cross-listed as PMATH 465

Formerly AM 474

AM 4385 W 3C 0.5 Applications of Algebra A survey of undergraduate mathematics with emphasis on the unifying effect of algebraic concepts. This is a cross-disciplinary course; theorems of Modern algebra are related to topics chosen mainly from advanced calculus and differential geometry. Prereq: MATH 234B or PMATH 336 Formerly AM 444

AM 441 Numerical Solution of Differential and Integral EquationsF 3C 0.5

Initial value problems: existence and uniqueness of solutions, one step methods, multistep methods, stability, error analysis. Boundary value problems: shooting and discretization methods, implementation problems especially for non-linear equations. Integral equations: correspondence io ordinary differential equations, initial value and boundary value problems, solution techniques.

Prereq: CS 374, or CS 337 and consent of instructor. CS 372 is recommended.

Cross-listed as CS 476

AM 451 * Introduction to Dynamical SystemsW 3C 0.5

A unified view of linear and nonlinear systems of ordinary differential equations in R". Flow operators and their classification: contractions, expansions, hyperbolic flows. Stable and unstable manifolds. Phase space analysis. Nonlinear systems, stability of equilibria and Lyapunov functions. The special case of flows in the plane, Poincaré-Bendixon theorem and limit cycles. Poincaré maps. Applications to physical problems will be a motivating influence.

Prereq: AM 351 or consent of instructor

Formerly AM 391

AM 453 Partial Differential Equations 2F 3C 0.5

A thorough discussion of the class of 2nd order linear partial differential equations with constant coefficients, in two independent variables. Laplace's equation, the wave equation and the heat equation in higher dimensions. Theoretical/Qualitative aspects: wellposed problems, maximum principles for elliptic and parabolic equations, continuous dependence results, uniqueness results (including consideration of unbounded domains), domain of dependence for hyperbolic equations. Solution procedures: elliptic equations - Green's functions, conformal mapping; hyperbolic equations - generalized D'Alembert solution, spherical means, method of descent; transform methods - Fourier, multiple Fourier, Laplace, Hankel (for all three types of partial differential equations); Duhamel's method for inhomogeneous hyperbolic and parabolic equations.

Prereq: AM 351 and 353, or consent of instructor

Formerly AM 481A

AM 455 Control TheoryW 3C 0.5

Dynamical systems, controllability and

observability. Minimization of functions

and functionals. Optimal control.

Pontryagin's Maximum Principle. Prereq: Consent of instructor Formerly AM 488

AM 456 Calculus of VariationsF 3C 0.5

Euler-Lagrange equations for constrained and unconstrained single and double integral variational problems. Parameter-invariant single integrals. General variational formula. The canonical formalism. Hilbert's independent integral. Hamilton-Jacobi. equation and the Caratheodory complete figure. Fields and the Legendre and Weier- Strass sufficient conditions.

Prereq: AM 231 or MATH 2308, or consent of instructor

Formerly AM 482

AM 463C Fluid Mechanics 1F 3C 0.5

Fundamental equations of inviscid fluids, compressibility, vorticity; two and three-dimensionat irrotational, incompressible flow. Blasius' theorem, Joukowski hypothesis. Water Wave Motion.

Prereq: AM 361

Formerly AM 466

Course Descriptions 16:101 Mathematics: Applied Mathematics

AM 464C Fluid Mechanics 2W 3C 0.5

Shock wave theory, supersonic flow around a corner, Prandtl-Meyer flow. Dynamics of real fluids, Navier-Stokes equations, exact solutions, Stokes and Oseen flow; Turbulence, Boundary layer theory. Introduction to Geophysical Fluid Dynamics.

Prereq: AM 463C

Formerly AM 476

AM 465C ElasticityW 3C 0.5

Basic equations of elasticity for homogeneous isotropic bodies; bending of beams; plane elastic waves; Rayleigh surface waves, Love waves. Solution of problems by potentials, variationat methods and Saint Venants' principle. Prereq: AM 361 Formerly AM 495

AM 473 Quantum Mechanics 2F 3C 0.5

Vector space formalism, Schrédinger and Heisenberg pictures, elements of second quantization. Angular momentum, selection rules, symmetry and conservation laws. Approximate methods: variation principle, perturbation theory and WKB approximation. identical particles, Pauli principle and simple applications to atomic, molecular, solid state, scattering and nuclear problems.

Prereq: AM 373 or consent of instructor

Formerly AM 4658

AM 475 Introduction to General RelativityF 3C 0.5

Equivalence principle. Curved spacetime and Einstein's gravitational field equations. The weak field limit. The Schwarzschild solution. Observational tests of General Relativity. Introduction to relativistic cosmology. Friedmann- Robertson-Walker universes and the big bang. Observational evidence. Gravitational collapse and black holes. Gravitational waves. Prereq: AM 375 or consent of

instructor

Formerly AM 372

Mathematics: Applied Mathematics

* 16:102