Applied Mathematics (AM)
41 courses in the 1990–91 Undergraduate Calendar, read from the scanned original. The text comes from OCR, so check the scanned page before relying on a detail.
AM 380B = =AM 481C
AM 395 AM 371C
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 soe
Listed without a description.
AM 466 AM 463C
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 430/40 - AM 4923/4 (not offered)
2. More detailed course descriptions and course outlines are available \.... the Applied Mathematics Handabook. -
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 func- _.. tions and their applications, including the role of uniform convergence.
Prereq: MATH 237
Antireq: MATH 210, 212, 213B, 2208, 230B, 240B
Formerly MATH 230B
This courses replaces MATH 2306 effective Winter, 1991.
A °S0 FW 3C 0.5 M elling with Ordinary Differential Equations ‘ Overview of the modelling process. E ples of physical systems leading to dinary differential equations. Appli- Cauuns to Newton's laws of motion, mechanical vibrations, and population dy ~amics. The emphasis is on the pt ical derivation and interpretation of th nodel equations.
Prereq: MATH 130B or 137
Antireq: MATH 215, 216
ormerly AM 260
Aw 251 F,W 3C 0.5 Modelling with Systems of Ordinary D/*-rential Equations S: lification procedures in modelling. P! _jical dimensions, scaling, and the principle of dimensional homogeneity. Buckingham's Pi Theorem. Systems of 0! ary differential equations. Phase p! 9» analysis. Stability, The empnasis is on two-dimensional physical systems based on the mechanics of pr~“sles and ecological systems. ‘ereq: MATH 138 or 230A, or
cc. ‘ant of instructor
Formerly AM 270
Al ‘52 F 3C 05 Fc sdations of Applied Mathematics Ar ut-depth look at some of the fundamental principles of applied mathem-“'4s, through case studies chosen fre classical mechanies of discrete ar :ontinuum systems and diffusion processes. Graup-theoretic foundations of dimensional analysis. Regular and sit ilar perturbation theory. req: AM 251 and one of MATH
24uA, 247 or AM 251 and one of MATH 230A, 237 plus consent of instructor
©+rmerly AM 280
Al 31 FW 3C 0.5 Real Analysis Topology of R", continuity, norms, mM 28, completeness. Fourier series ar applications, for example, to ordina, differential equations, the heat problem, optimal approximation, the iserarimetric inequality. req: MATH 230B or 237 1tireg: MATH 322A, PMATH 351 Cross-listed as PMATH 331 Not available for credit to students in H: urs Pure Mathematics programs.
Course Descriptions Applied Mathematics
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 integration, the Cauchy and allied theorems; Taylor and Laurent expansions, uniform convergence and power series; the residue calculus, and applications.
Prereq: MATH 230B 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 Ditterential 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-dimensionai 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, Laplace 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
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 2308 or 237. AM 250 is recommended for non-AM majors.
Formerly AM 381
16:9
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 elliptic 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 canonical 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 elasticity. Various applications.
Prereq: AM 231 or MATH 230B, 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 230B, or consent of instructor
Formerly AM 395
16:10 Course Descriptions Applied Mathematics
AM 373 Quantum Mechanics 1W 3C 0.5
Critical experiments and old quantum theory. Basic concepts of quantum mechanics: Observables, wavefunctions, Hamiltonians and the Schré- dinger equation. Uncertainty, correspondence and super-position principles. Simpte 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 Functional 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 Differential GeometryF 3C 0.5
Some global aspects of surface theory, the Euler-Poincaré characteristic, the global interpretation of Gaussian curvature via the Gauss-Bonnet formula. Submanifoids of £", 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 435 Applications of AigebraW 3C 0.5
A survey of undergraduate mathematics with emphasis on the unifying effect of algebraic concepts. This is a cross-disciplinary course; theorems ot 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 DifferentialF 3C 0.5
and integral Equations
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 to 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, expansion: hyperbolic flows. Stable and unstable manifolds. Phase space analysis. Nonlinear systems, stability of equilibrtand 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, ir two independent variables. Laplace's equation, the wave equation and the heat equation in higher dimensions. Theoretical/Qualitative aspects: weilposed 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 solutior spherical means, method of descent; transform methods - Fourier, multiple Fourier, Laplace, Hankel (for ail three types of partial differential equations); Duhamel's method for inhomogeneou: 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 anc
observability. Minimization of functior.
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 problams. 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 Weierstrass sufficient conditions.
Prereq: AM 231 or MATH 230B, 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-dimensional irrotational, incompressible flow. Blasius' theorem, Joukowski hypothesis. Water Wave Motion.
Prereq: AM 361
Formerly AM 466
AM 464C Fiuld 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 surtace waves, Love waves. Solution of problems by potentials, variational methods and Saint Venants' principle. ° Prereq: AM 361, or consent of instructor 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
Course Descriptions Applied Mathematics
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
AM 477 Statistical MechanicsF 3C 0.5
Equilibrium statistical mechanics is developed from first principles, based on elementary probability theory and quantum theory (classical statistical mechanics is developed later as an appropriate limiting case). Emphasis is placed on the intimate connections between statistical mechanics and thermodynamics. Although it would be useful, prior knowledge of quantum theory is not necessary. Prereq: Consent of instructor Formerly AM 486
AM 481C Applications of Information TheoryW 3C 0.5
Measures of information: probabilistic, mixed and nonprobabilistic. Measures of uncertainty (entropies). Entropies for continuous random variables. The maximum entropy principle and its applications to: statistical mechanics, optimal discretization and quantization of random variables, image representation and pattern recognition. Applications to questionnaires. Optimal questionnaires.
Prereq: Consent of instructor
Cross-listed as PMATH 380B
Formerly AM 380B
AM 491 Topics in Applied MathematicsF 3C 0.5
A selection of special topics given by members of the Applied Mathematics Department. Prereq: Consent of instructor Formerly AM 468
AM 492 Topics in Appiled MathematicsW 3C 0.5
A selection of special topics given by members of the Applied Mathematics Department. Prereq: Consent of instructor Formerly AM 478
AM 495 Reading Course Formerly AM 489° F 3C 0.5
16:11
AM 496 Reading Course Formerly AM 499W 3C 0.5
AM 493 Applications of Mathematics {Formerly AM 430}
Not offered in 1990–91.
AM 494 Applications of Mathematics {Formerly AM 440}
Not offered in 1990–91.
AM 359 Regional Planning: Program Development and ImplementationF 3C 0.5
1 examination’ of current regional
inning programs (objective, policies, -..ategies and plans), with regard to both their development and implementation in the context of various institu-
nal structures, arrangements and
ergovernmental relations. Emphasis will be given to the process of implementing and monitoring programs in “ferent jurisdictional and administra-
settings.
Prereq: Planning students or consent of instructor
-AN 360 W 3C 0.5
schnology in Urban and Regional Planning The influence of transportation, commu-
sations, and water and sewage
stems on the form, function and ~ velopment of cities and regions; the application of this knowledge in urban and regional planning.
-AN 367 W 3C 0.5 Conservation in Wildland and Resource Management ~ ynsideration of the constraints and
sidelines that an application of the ,-inciples of ecology place on the planning and management of resources within natural and agricultural ecosys-
ms. The theory and history of this ibject will be discussed together with the management of wildlife, forestry, fisheries, parks and open space, and jriculture. Prereq: ENV S 200 Cross-listed as GEOG 367 Lab fee $10-15 Students may receive credit for only ye of PLAN 367 and GEOG 367