38 courses in the 1986–87 Undergraduate Calendar, read from the scanned original. The text comes from OCR, so check the scanned page before relying on a detail.
Models for population growth, mixtures and diffusion, and motion of a point mass motivate the discussion of linear
and non-linear first and second order
differential equations. Prereq: MATH 130B Antireq: MATH 215, 216
Discussian of linear and non-linear systems of ordinary differential equations, including matrix methods, phaseplane analysis, stability; emphasis is on two-dimensional systems. Applications are drawn from mechanical systems in vibration, ecological systems, epidemics and planetary motion. Elements of dimensional analysis, scaling, and perturbation theory are introduced briefly. Prereq: AM 260, or MATH 2308, or consent of instructor Antiraq: AM 280
16.04 Course Descriptions Mathematics: Applied Mathematics
Case studies of typical problems in Applied Mathematics. Dimensional Analysis and the Pi theorem. Elementary group-theoretic methods. Scaling and perturbation theory.
Prereq: MATH 240A, or MATH 230A and consent of instructor.
Antireq: AM 270
Difference equations, Laplace transforms applied to discrete and continuous mathematical models taken from ecology, biciogy, economics and other fields. . Prereq: MATH 22086 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 2308 or consent of instructor Cross-listed as PMATH 366.
Stress and strain tensors; analysis of stress and strain. Lagrangian and Eulerian methods for describing flow. Equations of continuity, motion and énergy, constitutive equations. Navier-Stokes equation. Basic equations of elasticity. Various applications. Prereq: MATH 230B, AM 371, or consent of instructor : Coreq: MATH 382B (or PMATH 362)
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 260, 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 372
- with Applications
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.). Deter mination of practical measures of information.
Prereq: Consent of instructor
Cross-listed as PMATH 380A. -
Measures of conditional uncertainty (entropy). Maximizing conditional entropies. Applications ta 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
Crosslisted’ 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 2308 °
Systems of equations with constant or periodic coefficients. Floquet theory. Quasilinear systems. Stability analysis and Lyapunov functions. .
Prereq: AM 381.
“AM 464 W 3C 05.
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 2308 or consent ‘of instructor
Integral equations and integral iransforms will be applied to systems with memory. ‘
Prereq: Consent of instructor
AS a project, students will develop a
mathematical model and interpret its behaviour. ; : Prereq: Consent of instructor
A survey of undergraduate mathematics with emphasis on the unifying effect of — algebraic concepts. This is a cross- * disciplinary course; theorems of modem algebra are related to topics chosen mainly from advanced calculus and differential geometry.
Prereq: MATH 2348
Measure and integration The theory of measure and the Lebesgue integral.
Prereq: PMATH 331 or 351A
'
Topics in General Relativity
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 gravita-
tional waves. Other topics of current interest may be discussed.
Prereq: AM 362, 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
Prereq: Consent of instructor
Electronic properties of solids. Matrix mechanics. Symmetry and conservation laws. WKB method! Variation principle. Perturbation theory Bnd applications. Scatteting theory. Symmetry groups. Preraq: AM 465A
Fundamental equations of inviscid fiuids.
compressibility, vorticity, two and threedimensional irrotationdi, incompressible flow. Blasius’ theorem, Joukowski hypothesis. Water Wave Motion.
Prereq: AM 365
A selection of special tics given by members of the Applied atics Department.
Prereq: Consent of instructor
Linear operators in Hilbert spaces.
Compact operators. reduction to func.
tional analysis.
Prereq: AM 462 or dagger of instructor .
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Grassman aigebrag, differential fas, Lie derivatives, iscinetries, Killing vector fields, applications: to continuum impchanics difjererttiable manifolds and appications of differential forms. to physics.
Prereq: AM 362/PMATH 3685 or consent of instructor
Cross-listed as PMATH 465.
Shock wave theory, supersonic flow around a comer, Prandtl-Meyer: flow. Dynamics of real fluids, Navier-Stokes equations, exact solutions, Stokes and Oseen flow; Turbulence, Boundary layer theory. introduction to Geophysical Auid Dynamics. Prereq: AM 466
Prereq: Consent of instructor
Second-order partial differential equé- tions and characteristics; d'Alemberts’ solution of the wave equation, concepts
of distributions, construction of Green's -
functions, Fourier integral theorem, integral transforms, inverse transforms by contour integration. Applications to physical problems. Ja
Prereq: AM 371, 381, 391, or consent ©
of instructor
: i
AM A81B W 3C O58.
Partial Difterential Equations of - Applied Mathematics 3 Variational properties of eigenvalues and eigenfunctions, Rayleigh ‘Ritz method, special functions, Hankel transforms, Fourter-Bessel seride. sasymptotic series. Applications to Physibay problems. Prereq: AM 481A *
Euler-Lagrange tions! for - constrained ord ener ined single and double integral val rece problems. Parameter-invariant single integrals. “General variational forms The canonical formalism, ‘Hilbert’s pendent integral. Hamilton-Jacobf é%tation and the Caratheodory compete figure. Fields and the Legenre and Weier- _ strass sufficient conditions. : Préraq; MATH 2908 2 consent of instructor ea
Application’$ of dfaxwell’s equations. Introduction. to wave guides and Prereq: PHYS 253 or consent of instructor ..
- Equilibrium’ statistical mechanics is developed fram fist principles, based + On elementary prdability theory and quantum theory (classical statistical machari¢s 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
! -
Course Descriptions _ Mathematics: Applied Mathematics
16.95
Dynamical systems, controllability and ©
observability. Minimization of functions
and functionals. Optimal control.
Pontryagin’s Maximum Principle. Prereq: Consent of instructor
Basic equations of elasticity Yor 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 1986–87.
Not offered in 1986–87.
Not offered in 1986–87.