40 courses in the 1988–89 Undergraduate Calendar, read from the scanned original. The text comes from OCR, so check the scanned page before relying on a detail.
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
Simplification procedures in modelling. Physical dimensions and the principle of dimensional homogeneity. Buckingham’'s Pi Theorem. Scaling and introduction to perturbation theory. Systems of ordinary differential equations. Phase plane analysis. Stability. Emphasis on two and three dimensional physical systems.
Prereq: MATH 230A, or consent of instructor.
Case studies of typical problems in
applied mathematics. Classical mechanics of a point mass system. Conservative and dissipative systems. Existence and uniqueness theorems. Modelling with partial differential equations. The mathematics of diffusion processes. Group-theoretic foundations of dimensional analysis and scaling. Regular and singular perturbation theory. Prereq: AM 270, and MATH 240A (or MATH 230A and consent of instructor)
Difference equations, Laplace transforms applied to discrete and continuous mathematical models taken from ecology, biology, economics and other fields. Prereq: MATH 220B or consent of instructor
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: MATH 230B or consent of instructor
Cross-listed as PMATH 365
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: MATH 230B and AM 371, or consent of instructor
First order partial differential equations and characteristic curves; classification of linear second order partial differential equations; the Cauchy initial value
" problem; the Dirichlet problem. Solution
of the heat equation, wave equation, and Laplace's equation by separation of variables; Fourier series; propagation of
_ waves; the maximum principle for
harmonic functions. Prereq: AM 270 and MATH 2308, or consent of instructor
Flat space-time and Lorentz transformations, relativistic mechanics, Maxwell's equations; curved space-time and the Einstein field equations, the Schwarzschild solution and some experimental tests of general relativity, the weak field limit; introduction to black holes and cosmology.
Prereq: AM 362/PMATH 365 or consent of instructor
Antireq: AM 364
Variable length coding. The Shannon entropy as 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
Measures of conditional uncertainty (entropy). Maximizing conditional entropies. Applications to statistical physics, Statistics, pattern recognition. Special entropies in questionnaire theory. Design of optimal questionnaires. Special entropies in dynamical systems. Basics in dynamical systems. Prereq: Consent of instructor Cross-listed as PMATH 380B
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
Systems of equations with constant or periodic coefficients. Floquet theory. Quasi-linear systems. Stability analysis and Lyapunov functions.
Prereq: AM 381
Three-dimensional rigid body motion, Lagrange’s equations of motion. Ignorable co-ordinates. Hamilton's principle. Noether’s theorem and conservation laws. Canonical equations. Canonical transformations. Lagrange and Poisson bracket relations. Integral invariants. Prereq: MATH 230B or consent of instructor
A survey of undergraduate mathematics with emphasis on the unifying effect of algebraic concepts. This is a crossdisciplinary course; theorems of modern algebra are related to topics chosen mainly from advanced calculus and differential geometry.
Prereq: MATH 234B
The theory of measure and the Lebesgue integral. Prereq: PMATH 331 or 351A Cross-listed as PMATH 451
Topics chosen from: Charts and atlases, Manifolds and Diffeomorphisms, Tangent Spaces, Submanifolds, Vector Bundles, Tensor and Exterior Algebras, Differential Forms, Oriented Manifolds and Geometry, Homogeneous Spaces and Lie Groups.
Prereq: AM 362 or consent of instructor
Cross-listed as PMATH 463
Basic ideas of relativistic cosmology (Friedmann-Robertson-Walker universes, observational status of cosmological theories, black hole theory (the Schwarzschild and Kerr solutions, their event horizons and singularities, gravitational collapse, observable properties of black holes), and an introduction to gravitational waves. Other topics of current interest may be discussed.
Prereq: AM 362 and 372, or consent of instructor
Historical origins of quantum theory. Uncertainty principle, wave packets and the Schrédinger equation. The Hamiltonian operator. Solutions of onedimensional problems: harmonic oscillator, hydrogen atom, hydrogen molecular ion.
Prereq: Consent of instructor
Electronic properties of solids. Matrix
mechanics. Symmetry and conservation
laws. WKB method. Variation principle.
Perturbation theory and applications.
Scattering theory. Symmetry groups. Prereq: AM 465A
Fundamental equations of inviscid fluids.
compressibility, vorticity; two and three-
dimensional irrotational, incompressible
flow. Blasius’ theorem, Joukowski
hypothesis. Water Wave Motion. Prereq: AM 365
A selection of special topics given by members of the Applied Mathematics Department.
Prereq: Consent of instructor
Linear operators in Hilbert spaces. Compact operators. Introduction to functional analysis.
Prereq: AM 462 or consent of instructor
Cross-listed as PMATH 453
Course Descriptions 16:97 Mathematics: Applied Mathematics
Grassman algebras, differential forms, Lie derivatives, isometries, Killing vector fields, applications to continuum mechanics, differentiable manifolds and applications of differential forms to physics.
Prereq: AM 362/PMATH 365 or consent of instructor
Cross-listed as PMATH 465
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 466
A: selection of special topics given by members of the Applied Mathematics Department.
Prereq: Consent of instructor
Elliptic equations: ‘uniqueness theorems, Green’s functions and boundary value problems, potential theory, applications to electrodynamics and fluid mechanics, finite differences. Hyperbolic equations: Cauchy's problem, characteristics, Riemann’s methods, wave equation, applications to electrodynamics and fluid mechanics, finite differences.
Prereq: AM 371, 381 and 391, or consent of instructor
Parabolic equations: uniqueness theorems, Green's functions and boundary value problems, finite differences. Additional selected topics may include: Cauchy-Kowalewski Theorem, Hamilton- Jacobi Theory, maximum principles, formulation of well-posed problems, hyperbolic systems.
Prereq: AM 481A
16:98 Course Descriptions Mathematics: Applied Mathematics
Euler-Lagrange equations for constrained and unconstrained single and double integral variational problems. Parameterinvariant 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: MATH 230B or consent of instructor
Applications of Maxwell's equations. Introduction to wave guides and antennae.
Prereq: PHYS 253 or consent of instructor
Equilibrium statistical mechanics is devel-
oped 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
Dynamical systems, controllability and
observability. Minimization of functions
and functionals. Optimal control.
Pontryagin’s Maximum Principle. Prereq: Consent of instructor
Basic equations of elasticity for homoge-
neous 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 365
Not offered in 1988–89.
Not offered in 1988–89.
Not offered in 1988–89.
Not offered in 1988–89.