100 courses in the 1965–66 Undergraduate Calendar, read from the scanned original. The text comes from OCR, so check the scanned page before relying on a detail.
Notes
Faculty listed in 1965–66
R. G. Stanton, B.A. (Western), M.A., Ph.D. (Toronto) Professor and Chairman of the Department, Dean of the Faculty of Graduate Studies K. D. Fryer, B.A. (Western), M.A., Ph.D. (Toronto) Professor and Associate Chairman of the Department G. Berman, M.A., Ph.D. (Toronto) - - - - Professor H. Rund, B.Sc., Ph.D. (Cape Town), Dr. Habil. (Freiburg) Visiting Professor W. T. Tutte, M.Sc., M.A., Ph.D. (Cambridge) - - Professor G. E. Cross, M.A. (Dalhousie), Ph.D. (British Columbia) - Associate Professor H. F. Davis, S.M., Ph.D. (Massachusetts Institute of Technology) Associate Professor M. A. McKiernan, M.A. (Loyola), Ph.D. (Illinois Institute of Technology) Associate Professor
C. St. J. A. Nash-Williams, M.A., Ph.D. (Cambridge) Visiting Associate Professor
Department of Mathematics
Faculty listed in 1965–66
C. L. Siegfried, C.R., B.A. (Western), M.Sc. (Michigan) Associate Professor J R. A. Staal, M.A., Ph.D. (Toronto) - - Associate Professor D. G. Wertheim, B.A. (McMaster), M.A., Ph.D. (Toronto) Associate Professor C. F. A. Beaumont, B.A. (McMaster), M.A. (Toronto) - Assistant Professor A. Kerr-Lawson, B.A. (Toronto), S.M. (Chicago), Ph.D. (McMaster) (on leave of absence, 1964-65) Assistant Professor R. C. Mullin, B.A. (Western), M.A., Ph.D. (Waterloo) - Assistant Professor P. J. Ponzo, B.A.Sc., M.A. (Toronto), Ph.D. (Illinois) - Assistant Professor K. A. Rowe, B.A. (Toronto), M.S. (Wisconsin).
Ph.D. (Illinois) v. - - - - Assistant Professor E. R. Bishop, B.Sc. (Acadia), M.Sc. (Queen’s) - - Lecturer T. M. K. Davison, B.Sc. (Sir George Williams), M.A. (Toronto) - Lecturer R. A. Honsberger, B.A. (Toronto) - - - - Lecturer W. P. Kotorynski, B.Sc. (Western), M.A. (Toronto) - Lecturer R. B. Reed, B.A.Sc. (Toronto), M.Sc. (Waterloo) - - Lecturer R. P. Rigelhof, B.E. (Saskatchewan), M.Sc. (Waterloo) - Lecturer G. N. Robertson, M.Sc. (Manitoba) - - Lecturer (part-time) P. L. J. Ryall, B.Sc. (London), M.A. (Cambridge) - - Lecturer K. Salkauskas, B.A.Sc. (Toronto), M.Sc. (Waterloo) - - Lecturer
Computer Science Division
J. W. Graham, M.A. (Toronto) Associate Professor and Director of Computing Centre
Faculty listed in 1965–66
J. D. Lawson, B.A.Sc. (Toronto), M.Sc. (Waterloo)
(on leave of absence, 1964-65) Assistant Professor D. D. Cowan, B.A.Sc. (Toronto), M.Sc. (Waterloo) - Lecturer P. W. Shantz, B.Sc., M.A. (Waterloo) - - - - Lecturer
Statistics Division
Faculty listed in 1965–66
G. Menges, Dr.rer.pol. (Frankfurt) ‘. - Visiting Professor D. A. Sprott, M.A., Ph.D. (Toronto) - - - - Professor M. Behara, M.Sc. (Utkal), Dr.rer.oec. (Saarland) - Assistant Professor G. W. Bennett, B.Sc., Ph.D. (Adelaide) - - Assistant Professor J. G. Kalbfleisch, B.Sc. (Toronto), M.Sc. (Waterloo) - Lecturer
Functions and limits, the derivative. Differentiation formulae. Applications to tangents, rates, extrema. The indefinite and definite integrals, fundamental theorem of integral calculus. Applications to area, volume, centroids, moments of inertia, fluid pressure, work, potential. Introduction to the trigonometric, inverse trigonometric, exponential, logarithmic functions. Transcendental functions. Parametric and polar equations. Formal integration and applications to physical problems. Students will work selected problems under supervision. A certain proportion of the problems will include mathematical formulation of physical problems.
The rational, real, and complex number systems. Polynomial functions. Theory of equations. Determinants. Vector and matric notation. Co-ordinates in three dimensions. Systems of planes and lines. Standard equations of quadric surfaces. Spherical and cylindrical co-ordinates. 2 lectures, 1 hour problems.
A continuation of course 12. Infinite series and power series. Partial derivatives. Multiple integration and applications to areas, volumes, centroids, moments. Introduction to differential equations.
Introduction to the use of desk calculators. Simple problems on interpolation, solution of algebraic equations, numerical evaluation of integrals, systems of equations.
Standard methods of solution. Applications to physical and engineering problems. Linear equations with constant co-efficients. Systems of differential equations. Solution by series. Numerical methods of solution. Partial differential equations; applications of partial differential equations.
Prerequisite: Math 22.
3 lectures, one term.
32, Numerical Analysis. Ordinary, divided, and central difference interpolation formulae. Graphic and numerical solutions of algebraic and transcendental equations. Computation with series and integrals. Systems of linear equations. Determinants and matrices. Eigenvalue problems. Iterative procedures. Relaxation methods. Difference equations. Mathematical principles associated with electronic computers.
Real numbers, sequences, limits, continuity. The derivative. General Theorem of Mean Value. Functions of several variables.
Department of Mathematics
implicit functions, Jacobians. Power series with complex terms, the Taylor series for functions of several variables, constrained extrema. The elementary functions for a complex variable.
Prerequisites: Math 22, 31.
3 lectures, one term.
34, Integral Calculus. Systematic treatment of formal integration. Differential equations. Calculus of variations. Fourier series. Riemann integration, improper integrals, multiple integrals with applications. Line integrals, theorems of Green and Stokes.
Prerequisites: Math 22, 31.
3 lectures, one term.
41, Applied Analysis I. Partial differentiation, Laplace transforms, special functions. Fourier series, second order differential equations. 3 lectures, one term.
Scalar and vector products; curl, divergence, and gradient. Use of vectors and matrices in discussion of physical problems in mechanics, fluid flow, and electromagnetism. Introduction to tensors. Prerequisites: Math 22, 31, and consent of instructor.
Cauchy-Riemann equations, the Cauchy integral theorems, conformal mapping, the Taylor and Laurent series, contour integration. Prerequisite: Math 33, 34.
Frequency distributions; probability; Binomial, Normal, and Poisson laws. Sampling theory. Distribution of Chi-squared, F, and t. Hypothesis testing, quality control, regression theory. Curve fitting. Analysis of variance, introduction to experimental design.
The Heaviside Calculus and the Laplace Transform. Admittance, impedence, and other operators. Applications of operator techniques to problems in electricity, mechanics, and heat.
Prerequisites: Math 22, 31, and consent of instructor.
MATH 54Differential Equations of Mathematical Physics
The wave equation. Potentials. Boundary-value problems. Eigenvalue problems. Expansions in series of orthogonal functions. Spherical, cylindrical, and ellipsoidal harmonics. Prerequisites: Math 22, 31, and consent of instructor.
Fundamentals of digital computers, in cluding memory types, arithmetic components, input-output and control units. Binary numbers, floating-point arithmetic, flow charting, machine-language coding. Automatic coding, including assemblers, compilers, and translators. Interpretive routines.
Cartesian Co-ordinates. The distance formula. Functions and their graphical representation. The linear function. Forms of the line equation. The quadratic function and roots of quadratic equations. Algebraic and geometric approach to loci. Standard forms for the circle, parabola, ellipse, hyperbola. The number system. Polynomial functions. Remainder theorem. Functions of more than one variable. Equations in one or more variables. The general trigonometric functions, their values and graphs. Radian measure. Compound angle formulae. Trigonometric formulae. Trigonometric equations. The sine and cosine laws for triangles. The logarithmic function. The laws of logarithms. Arithmetic and geometric series. The binomial series. Properties of the binomial coefficients. Permutations and combinations.
A terminal course designed for Arts students who wish to learn something of the fundamental ideas of mathematics but who will not be encountering technical problems of application. The course will include discussion of the fundamental concepts involved in a selection of topics chosen from mathematical areas such as Calculus, Statistics, Probability, Projective Geometry, Non-Euclidean Geometry, Set Theory, Group Theory, Field Theory, Boolean Algebra, Mathematical Logic, Vector Spaces, Matrices, Numerical Analysis.
Functions and limits. The derivative. Dfferentiation of algebraic functions. Applications to tangents, rates, minima and maxima. Integration as the limit of a sum. Fundamental theorem of integral calculus with applications to areas, volumes, and moments. Power series. Partial differentiation. 3 lectures.
The real and complex number systems. Polynomial functions. Theory of equations. Determinants. Vector and matrix notation. Elementary solid geometry, planes and lines, the standard quadric surfaces. Co-ordinate systems. 2 lectures, one hour problems.
The topics of Mathematics 130, presented with a view to the special needs of Social Science students. Emphasis will be on basic concepts rather than on the more difficult manipulative techniques.
Real numbers, sequences. Limits. Continuity. Derivatives. Functions of several variables. Implicit functions. Jacobians. Taylor’s series with geometric applications. Power series with complex terms. The elementary functions for a complex variable. °
Methods of formal integration. Differential equations. Fourier series. Riemann integration and applications. Calculus of variations. Line integrals. Stokes’ and Green’s theorems.
Frequency distributions. Probability. Binomial, Normal, and Poisson laws. Sampling theory. Curve fitting. Distributions of Chisquared, F, and t. Hypothesis testing. Quality control. Regression theory. Analysis of variance. Introduction to experimental design.
The statics of particles and rigid bodies, work and the stability of equilibrium. Motion of a particle in a uniform field. Oscillatory motion in. one dimension. Motion of a system of particles and of a rigid body. Central forces. Accelerated reference systems. Motion of a rigid body in three dimensions.
An introduction to first order differential equations. Applications to growth, decay, reactions. Linear equations
187
a’
with constant coefficients. Systems of equations; vector and matric methods. Solutions in series. Introduction to partial differential equations. 2 lectures.
The topics of Mathematics 233 presented with a view to the special needs of Social Science students. Emphasis will be on basic concepts rather than on the more difficult manipulative tecbniques.
The topics of Mathematics 243 treated without the use of Calculus. In addition to the basic statistical methods, emphasis will be placed on the nature of evidence and inference.
The nine-point circle. The Simson line. Cross ratio. Projection. Harmonic ranges and involutions. Properties of circles. Axioms of incidence for projective geometry. Duality. Theorems of Pappus and Desargues. Introduction of co-ordinates. Conic sections. Introduction to non-Euclidean geometry.
Vectors and vector spaces; transformations and matrices; quadratic and Hermitian forms. Algebraic structures; rings and ideals. Matrices with entries from a ring. Canonical forms and eigenvalues; the orthogonal group.
The real number system. Uniform convergence. Theory of Fourier series. The complex number system. Analytic functions. The Cauchy theorems. Contour integration. Singularities. Conformal mapping. 3 lectures.
First-order differential equations. Linear equations with constant coefficients. Solutions in series. Systems of differential equations. Introduction to partial differential equations. Introduction to integral equations.
Finite differences. Solution of equations. Computation with series and integrals. Linear systems and matric methods. Difference equations and relaxation methods. Numerical solution of partial differential equations. Methods employing electronic computers.
A theoretical course in the calculus of finite differences to include summation, the differences of zero, numerical integration, the relation between integration and summation. An introduction to difference equations. 2 lectures.
An advanced mathematical course including the the multivariate normal law and related distributions. Normal regression theory. Applications to hypothesis testing and analysis of variance.
Introduction to basic concepts and theorems, connectivity, planarity, arboricity, directed graphs, partial orderings, girth and regularity, the four-colour problem.
Topological concepts. Analytic continuation. The Riemann surface. Entire meromorphic functions. Periodic functions. Special functions. General metric spaces. Function spaces. The integral as an operator.
Theory of metric spaces, completeness and compactness, measure in Euclidean n-space, the Lebesque integral, convergence theorems, the Fubini thoerem, differentiability, absolute continuity. A study of Banach spaces.
MATH 434Differential Equations of Mathematical Physics
Boundary-value problems. Orthogonal functions. Bessel functions. The Legendre polynomials. The Fourier integral. The Laplace transform. The wave equation.
An informal introduction to the logic of sentences and predicates, with emphasis on analogies with familiar mathematical structures. Syllogisms and the algebra of sets related to predicate logic. Simplification, logical deduction, duality, consistency and completeness. The concepts of constant, ‘variable, function, and set. Axiomatics.
The mathematics and logic of estimation and hypothesis testing. Information. Consistency. Efficiency. Sufficiency. Fiducial and confidence intervals. The problems of large and small samples. 2 lectures.
439, Theory of Experimental Design. The logic of experimental design. Randomization. Experimental and sampling errors. Mathematical methods of constructing various designs. Applications to problems in the sciences.
Recurrent events. Markov processes and applications to Physics and Biology, including diffusion processes and epidemics. 2 lectures.
441, Quantum Mechanics. The Schroedinger wave equation, interpretation of wave functions, harmonic oscillator in wave mechanics. The hydrogen atom. Perturbation theory. The variation methods. The W.K.B. approximation.
2 lectures.
442, Theory of Relativity. Covariant and contravariant tensors. Riemannian space. Gaussian curvature. The Lorentz transformation and Einstein’s special theory. Introduction to the general theory of relativity.
The development of Mathematics from ancient to modern times, including study of prominent mathematicians and their works. Problems will be worked using both modern and historical approaches. 2 lectures.
Logic of experimental designs. Randomization. Experimental and sampling errors. Designs applied in biology. Methods of analysis for factorial designs, Latin squares, etc.
Applied Mathematics for Science majors. Ordinary and partial differential equations. Laplace and Fourier transforms. Special functions. Fourier series. Vector and matric methods. Calculus of variations. Other topics in advanced calculus.
2 lectures.
Graduate and Research Programmes
While the department offers graduate work over a broad spectrum of the various branches of Mathematics, fields of special interest include: Abstract Algebra, Combinatorial Analysis, Computer Science, Decision Theory, Differential Equations, Differential Geometry, Foundations, Functional Analysis, Functional Equations, Graph Theory, Harmonic Analysis, Mathematical Logic, Number theory, Numerical Analysis, Operations Research, Probability, Statistical Inference.
Extensive computer facilities are available in the Computer Centre and include an IBM 1401, an IBM 1620, an IBM 1710 system, and an IBM 7040. Each year, one or more Visiting Professors spend a term with the Department. Recent visitors include: G. Temple (Oxford), W. B. Pennington (Aberystwyth), H. Rund (South Africa), G. Menges (Saarbrucken), C. St. J. Nash- Williams (Aberdeen).
Extensive opportunities for fellowship assistance are available through National Research Council Fellowships and Province of Ontario Graduate Fellowships. In addition, the Department of Mathematics offers a number of Teaching Fellowships which may be held singly, or in conjunction with other Fellowships. For details, correspondence should be addressed to the Department Chairman.
Candidates for the degrees of M.A., M.Sc., M.Phil. and Ph.D. are accepted under the general regulations set forth in the section of the calendar pertaining to the Faculty of Graduate Studies. Candidates for the M.A. or M.Sc. degrees in Mathematics are normally advised to take the degree by course work rather than by thesis.
Concepts from thermodynamics. One-dimensional gas dynamics. One-dimensional wave motion. Waves in supersonic flow. Flow in ducts and wind tunnels. Methods of measurement. Equations of frictionless flow. Small-perturbation theory. Bodies of evolution, and slender body theory. The similarity rules of high-speed flow. Transonic flow. The method of characteristics. Effects of viscosity and conductivity. Laminar and turbulent boundary layers. Concepts from gas kinetics.
604, Special Functions. Elliptic functions. The hypergeometric function. The gamma and beta functions. The functions of Laplace, Legendre, Lamé, Mathieu, and Bessel. Orthogonal functions; the functions of Laguerre, Hermite, Chebyshev.
(a) Introduction to Random Variables. Probability, random variables, distribution functions, expectation, discrete transform methods. 1 term.
(b) Theory of Probability. Measures on a sample space, types of convergence, limit theorems, laws of large numbers.
1 term.
({c) Stochastic Processes I. Discrete space — discrete time processes. Discrete space — continuous time processes. Applications in Physics, Biology, Queuing Theory, etc.
1 term.
(d) Stochastic Processes II. Mathematical foundations of stochastic processes, continuous space processes. Time-series analysis.
MATH 614A series of courses in the applications of Analysis
(a) Integral Equations 1 term.
Department of Mathematics
(b) Integral Transforms 1 term. (c) Complex Variable Techniques 1 term. (@) Fourier Series and Orthogonal Functions 1 term. (e) Group Theory in Physical Problems 1 term.
General concepts of computer types and applications. Flow charting, Cqding, instruction formats. Editing. Storage elements and buffers. Control and accuracy in digital computers. Input and output devices and their use. Number systems. Automatic programming and universal computer languages. Logic and Boolean Algebra. Heuristic programming. Elements of logical design.
Vector spaces, linear transformations, matrices. Euclidean spaces, groups of transformations, algebra of tensors; covariant differentiation, divergence, Laplacian, curl. Generalizations to Riemannian spaces. Applications to other fields including differential geometry, mechanics, elasticity, hydrodynamics, electricity, and general relativity.
Non-numeric computing. List processors and their application to the calculation of recursive functions, language translation, analytic differentiation and integration, mathematical logic, sorting, heuristic programming. Polish strings. Simulation of a List Processor.
MATH 641Mathematical Methods for Operations Research
A series of courses with
emphasis on mathematical techniques having direct application to business.
industrial, military, and scientific problems.
(a) Special Methods. Algebra of probability distributions; Monte Carlo methods; application to queuing theory, replacement theory, and inventory control.
1 term.
(b) Model Analysis. Partially ordered models. stochastic models, general model logic; examples from business, industry, and the military including the Leontiey model from Economics; construction of models, optimization, computational methods.
1 term.
(c) Competitive Models. A selection of topics from game theory, gaming, and decision theory. 1 term.
(a) Analytic Methods. Review of classical methods; optimization of discrete functions, inequalities, introduction to linear programming, steepest descent and optimal control.
(c) Linear Programming. Simplex method; flows in networks; application to transportation problems; assignment problems, and project cost curves. 1 term.
(d) Programming. A selection of topics from non-linear programming including the simplex method for quadratic programming, Lagrangian differential, gradiant and gradient projection; introduction to integer programming and dynamic programming.
The Euler equations and the Lengendre and Jacobi conditions. The Weierstrass E function and sufficient conditions. The theory of transversals and isoperimetric problems. Hilbert’s existence theorem. Selected topics from the theories of Kneser and Caratheodory, the problem of Bolza.
G. T. Csanady, Dipl. Ing. (Munich), Ph.D. (New South Wales) Professor and Chairman of Department
S. Alpay, Dipl. Ing., Dr. Ing. (Berlin) - - - - Professor W. B. Heginbotham, B.Sc., Ph.D. (Manchester) - - Visiting Professor H. L. Evans, M.Sc. (Wales), D.I.C., Ph.D. (London) - Associate Professor T. A. Brzustowski, B.A.Sc. (Toronto), A.M., Ph.D. (Princeton) Associate Professor M. J. Hillier, B.Sc. (Eng.), B.Sc. (Gen) (London), D.1.C., M.Sc. (Eng.) (London) - - - Associate Professor
J. H. G. Howard, B.Sc. (Queen’s), M.Sc., Ph.D. (Birmingham) Associate Professor
W. B. Nicoll, S.M. (Massachusetts Institute of Technology) _ Engineer (Stanford) - - - - - Associate Professor
G. F. Pearce, B.A.Sc. (British Columbia), M.A.Sc. (Toronto) Associate Professor
A. A. Bruneau, B.A.Sc. (Toronto), D.LC., Ph.D. (London) Associate Professor
J. W. Church, B.Sc. (Queen’s), M.A.Sc. (Toronto) - Associate Professor D. E. Coates, B.A.Sc. (Toronto), M.S. (Illinois) - Assistant Professor
G. D. McPherson, B.A.Sc. (Toronto), M.S. (Ottawa) Ingenieur en Genie Atomique (Institut National des Sciences et Techniques
Nucleaires, Saclay, France) - Assistant Professor (on leave of absence) A. Plumtree, B.Sc., Ph.D. (Nottingham) - - Assistant Professor K. R. Piekarski, Dipl. Ing. (London) - - Assistant Professor G. K. Fleming, B.Eng. (Nova Scotia Tech. College).
M.A.Sc. (British Columbia) - - - - Lecturer V. Kuppu Rao, B.E. (Bangalore), M.S. (Cornell) - - Lecturer A. B. Strong, B.A.Sc. (Waterloo) - - - - Lecturer
Undergraduate Programmes
Details of the undergraduate programme in Mechanical Engineering are to be found on page 68. All courses extend over one semester only.
A basic course in kinematics emphasizing the creative approach. Uniform rotary motion including rolling cylinders, gears and the synthesis of gear trains including planetaries. Non-uniform motion including cams, gears and linkages. Programming non-uniform motion and the synthesis of cam mechanisms, non-circular gear trains and linkages.
The effects of dynamic forces in machines; introduction to mechanical vibrations. Transient forces. Static force analysis, dynamic force analysis, balancing, cam dynamics, dynamics of feedback systems. Prerequisite: ME13.
Theory and application of mechanics of solids to machine parts. Design of machine elements; power screws, shafting, bearings, belts, gears, flywheels, clutches and brakes, crankshafts, springs, lubrication, ball and roller bearings.
Principles of optimum design, optimizing for minimum cost, maximum power capability, minimum weight, effects of manufacturing errors on product performance, statistical consideration for the factor of safety, optimum design of machine elements; torsion shafts, beams, gears, etc. Fluctuating loads, impact forces. Analog methods. Prerequisite: ME 22.
A broad development of mechanical vibration and impact. Transient phenomena, self-excited vibration, torsional oscillations, multi-mass systems, electromechanical systems and analogies. Forced vibrations, elimination and absorption of vibration. Measurement of vibration.