38 courses in the 1991–92 Undergraduate Calendar, read from the scanned original. The text comes from OCR, so check the scanned page before relying on a detail.
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, 220B, 2308, 240B
Formerly MATH 230B
This courses replaces MATH 230B effective Winter, 1991.
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
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
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. Prereq: 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.
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 2308 or 237
Antireq: MATH 322B, 332B, PMATH 352
Cross-listed as PMATH 332
Formerly MATH 3328
Not available for credit to students in Honours Pure Mathematics programs.
the Serret-Frenet formulae; surfaces in ES 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 230B, or consent of instructor
Cross-listed as PMATH 365
Formerly AM 362
Pitfalls in computation; solution of linear algebraic equations, polynomial interpolation; least squares; numerical integration and differentiation. The intent is to expose students to the theory behind modern algorithms for solving mathematical problems.
Prereq: CS 132 or 140, MATH 134B or 136, MATH 235 or (MATH 230B and 234A) is recommended.
Antireq: CS 372, 374
Cross-listed as CS 337
Course Descriptions Applied Mathematics
trans@rms 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
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
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
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 2308, or consent of instructor
Coreq: AM 353 and AM/PMATH 332 (or PMATH 352}
Formerly AM 365
16:9
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
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. 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
transformations. Physical consequences. Optics including Doppler effect and observation. Relativistic mechanics. Collisions. E = me?. 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
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-listad as PMATH 380A
Formerly AM 380A
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
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
Some global aspects of surface theory, the Euler-Poincaré characteristic, the global interpretation of Gaussian curvature via the Gauss-Bonnet formula. Submanifolds of E", induced Riemanhian 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 appli- Cations to continuum mechanics and mathematical physics.
Prereq: AM 333/PMATH 365 or consent of instructor
Cross-listed as PMATH 465
Formerly AM 474
Course Descriptions Applied Mathematics
A survey of undergraduate mathematics with emphasis on the unifying effect of algebraic concepts. This @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
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
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
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
Dynamical systems, controllability and
observability. Minimization of functions
and functionals. Optimal control.
Pontryagin’s Maximum Principle. Prereq: Consent of instructor Formerly AM 488
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 Weierstrass sufficient conditions.
Prereq: AM 231 or MATH 230B, or consent of instructor
Formerly AM 482
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
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
Basic equations of elasticity for homogeneous isotropic bodies; bending of beams; plane elastic waves; Rayleigh surface waves, Love waves. Solution of problems by potentials, variational methods and Saint Venants' principle. Prereq: AM 361, or consent of instructor Formerly AM 495
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, sglid state, scattering and nuclear problems.
Prereq: AM 373 or consent of instructor
Formerly AM 465B
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
Course Descriptions Applied Mathematics
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
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
A selection of special topics given by members of the Applied Mathematics Department. Prereq: Consent of instructor Formerly AM 468
A selection of special topics given by members of the Applied Mathematics Department. Prereq: Consent of instructor Formerly AM 478
Not offered in 1991–92.
Not offered in 1991–92.
Not offered in 1991–92.