MATH 360(a) Tensor Analysis

1972–73 to 1974–75 3 calendars

Elementary introduction to classical differential geometry and tensor analysis. Curves and surfaces in Euclidean spaces, coordinate transformation, invariance of length, introduction of tensors and metric, Riemannian spaces, covariant differentiation, geodesics, curvature and Einstein tensors. 2 hours lectures, + 1 hour tutorial, Fall term Prerequisites: M237, consent of instructor

• 1972–73 PDF p. 438
• 1973–74 PDF p. 472
• 1974–75 PDF p. 532

1968–69 to 1971–72 4 calendars

Vector analysis in curvilinear coordinates; gradient, divergence, curl. Transformation laws of covariant, contravariant, relative tensors. Covariant and intrinsic differentiation. The Riemann-Christoffel tensors, Bianchi identities, Ricci and Einstein tensors. Integrability theorems. Applications to classical mechanics and elasticity. Introduction to general relativity.

2-hours lectures, Fall term.

(b) Differential Geometry. The metric tensor, and elementary differential geometry of surfaces, Introduction to Riemannian geometry; Riemannian curvature, Gaussian curvature and Einstien curvature tensors. Spaces of affine connection and Finsler spaces with application to electromagnetic theory.

2 hours lectures, Winter term.

• 1968–69 PDF p. 280
• 1969–70 PDF p. 266
• 1970–71 PDF p. 328
• 1971–72 PDF p. 357

1967–68 1 calendar

Another title: MODEL 40. The Department of Electrical Engineering operates an, probably a different course under the same number (or a misread scan).

analogue computer in conjunction with the 1710 to give a hybrid system.

The IBM 1620 computer is available to undergraduate students at all times. The IBM 1710 is used by undergraduates, graduates, faculty and staff according to schedules made up one week in advance. Both the 7040 and the Systems/360 are run on a closed shop basis, but all members of the University can get programmes processed within a reasonably short time.

Systems planned for the future include an IBM System/360, Model 44, and, at a later date, an even more powerful Systems/360 Computer.

Extensive opportunities for fellowship assistance are available through National Research Council Fellowships and Province of Ontario Graduate Fellowships. In addition, the Faculty 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. Math., M. Phil. and Ph.D. are accepted under the general regulations set forth in the section of the calendar pertaining to the Graduate Studies.

PURE MATHEMATICS: Current research in pure mathematics includes studies in functional equations for analytic functions, categories of incidence geometries, theory of integration and measure, the enumeration of planar maps, matrixlike representations of infinite groups, homotopy theory, contact manifolds, theory of generalized decompositions of groups, logic and foundations of set-theory. and general functional equations.

Functional equations — that is, roughly speaking, the theory of determining

functions from elementary equations containing them — is a field of mathematics looking back into a history of more than two centuries. the somewhat general theory of which has been developed however only in the last two decades and it has applications in and is connected with such modern disciplines as the theory of information, mathematical psychology and universal algebra.

At the University of Waterloo we work on these problems, on general methods and uniqueness theorems, and also on applications to and relations with differential geometry, homological algebra, functional analysis, theories of continuous groups of nets and of complex functions. etc.

COMPUTER SCIENCE: Research in compiler construction has produced the WATFOR fast FORTRAN IV compiler, completed in 1965 and currently used by over seventy 7040-7044 computing centres throughout the world. Research and development work on similar compilers for the 360 series machines is progressing well.

A graphic display unit will permit research in new techniques in computer-aided design, computer graphics, graphic programming and simulation. Also, a process control computer is currently used to perform and analyze complex experiments, some of which wouid be difficult to handle by conventional methods. In numerical analysis, algorithms for two-point boundary problems, stiff systems of differential equations, unilateral matrix equations and optimization problems are under investigation.

In addition, many computer-oriented research projects by faculty and graduate students in science, engineering and the humanities are pursued with the assistance of the Computer Science department. It is this wide variety of problems which makes the department environment so stimulating to faculty and students alike.

STATISTICS: Research in the Department of Statistics centres on the controversial problem of formulating exact statistical inferences in many diverse situations using generalizations of the approach introduced by the late Sir Ronald Fisher in his foundation of the theory of fiducial inference, the theory of the likelihood function and the theory of estimation in general. The extensive computing facilities available at the University of Waterloo have given rise to investigations into the uses of computers to obtain exact results on the basis of these theories in situations previously treated by asymptotic methods, and also research into exact tests in multivariate analysis.

Closely related to this research, there is active interest in problems of experimental design, with particular reference to incomplete designs, multivariate analysis, estimation theory, mathematical genetics and stochastic processes.

Members of the department are also interested in problems in combinatorial mathematics, such as the construction of balanced incomplete block designs, which arise in the design of experiments. Other combinatorial questions under consideration include edge-chromatic graphs and the Ramsey numbers.

Faculty of Mathematics

Increasing contact with the departments of Chemistry, Physics and Psychology provides a variety of new problems and also gives advanced students a valuable opportunity to apply lecture material to useful practical situations. An arrangement with the Statistics Unit of the Department of Agriculture in Ottawa provides potential summer employment for advanced undergraduates and at the same time, a further opportunity for practical experience.

APPLIED MATHEMATICS: Research in this department is being carried out in a number of areas directly or indirectly applicable to problems in the Physical Sciences. The range of interests of the members of the department reflects the diversity of problems of a mathmatical nature which arise in the real world.

The major areas of specialization include Nonlinear Ordinary Differential Equations, Spectral Theory of Differential Operators, Partial Differential Equations, Potential Theory, Functional Equations, Calculus of Variations, Boundarylayer Problems in Fluid Dynamics, Harmonic Analysis and Differential Geometry.

Graduate Course Descriptions

§50. Axiomatic Set Theory. First order theories and their properties. Formal number theory. Set theory as a first order theory. Elimination of the logical paradoxes. Comparison of several axiom systems for set theory. Basic concepts of mathematics. Bourbaki’s structure theory and the theory of categories. The theory of ordinals. Transfinite arithmetic. The axiom of choice. The axiom of restriction. The generalized continuum hypothesis. Summary of recent independence proofs.

• 1967–68 PDF p. 258