AM 250 F,W,S 3C,1T 0.5
Introduction to Differential Equations
Physical systems which lead to differential equations (examples include mechanical vibrations, population dynamics, and mixing processes). Dimensional analysis and dimensionless variables. Solving linear differential equations: first- and second-order scalar equations and first-order vector equations. Laplace transform methods of solving differential equations.
Prereq: MATH 138/108
Antireq: MATH 218, 228, AM 251
AM 261 F 3C,1T 0.5
Newtonian Mechanics
Modelling physical reality: Mathematics vs. Physics. Point-particle model. Kinematics. Dynamics: Newton's laws of motion. Critique of Newton's formulation. The principle of galilean invariance. Applications: standard problems of particle motion. The conservation principles: energy, linear momentum, and angular momentum. Collision processes. The two-body problem with a central field. Introduction to the linear theory of small oscillations: normal co-ordinates, weak-coupling limit, forced vibrations of two coupled degrees of freedom.
Prereq: MATH 237
AM300s
AM 331 F,W 3C 0.5
Real Analysis
Topology of Rn, continuity, norms, metrics, completeness. Fourier series and applications, for example, to ordinary differential equations, the heat problem, optimal approximation, the isoperimetric inequality.
Prereq: MATH 237
Antireq: PMATH 351
Cross-listed as PMATH 331
Not available for credit to students in Honours Pure Mathematics programs.
AM 332 W,S 3C 0.5
Complex Analysis
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 237
Antireq: PMATH 352
Cross-listed as PMATH 332
Not available for credit to students in Honours Pure Mathematics programs.
AM 333 F 3C 0.5
Elementary Differential Geometry and Tensor Analysis
Curves in Euclidean 3-Space (E3) and the Serret-Frenet formulae; surfaces in E3 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 consent of instructor
Cross-listed as PMATH 365
AM 343 W 3C 0.5
Discrete Models in Applied Mathematics
Difference equations, Laplace and z transforms applied to discrete (and continuous) mathematical models taken from ecology, biology, economics and other fields.
Prereq: MATH 108, or consent of instructor
AM 351 F,S 3C 0.5
Ordinary Differential Equations
Existence and uniqueness theorems; first-order and second-order 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 237, AM 250 is recommended for non-AM majors
AM 353 W,S 3C 0.5
Partial Differential Equations 1
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, or consent of instructor
Coreq: AM 351
AM 361 W 3C 0.5
Continuum Mechanics
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 and AM 261, or consent of instructor
Coreq: AM 353 and AM/PMATH 332 (or PMATH 352)
AM 373 W 3C 0.5
Quantum Mechanics 1
Critical experiments and old quantum theory. Basic concepts of quantum mechanics: observables, wavefunctions, hamiltonians and the Schrödinger equation. Uncertainty, correspondence and superposition principles. Simple applications to finite and extended one-dimensional systems, harmonic oscillator, rigid rotor and hydrogen atom.
Prereq: AM 231 and AM 261, or consent of instructor
AM400s
AM 431 F 3C 0.5
Measure and Integration
General measures, measurability, Caratheodory Extension theorem and construction of measures, integration theory, convergence theorems, Lp-spaces, absolute continuity, differentiation of monotone functions, Radon-Nikodym theorem, product measures, Fubini's theorem, signed measures, Urysohn's lemma, Riesz Representation theorems for classical Banach spaces.
Prereq: PMATH 354 or consent of instructor
Cross-listed as PMATH 451
AM 432 W 3C 0.5
Functional Analysis
Banach and Hilbert spaces, bounded linear maps, Hahn-Banach theorem, Open Mapping Theorem, Dual spaces, weak topologies, Tychonoff's theorem, Banach-Alaoglu theorem, reflexive spaces, compact operators, Spectral theorem, commutative Banach algebras.
Prereq: PMATH 354 or consent of instructor; AM 431/PMATH 451 is recommended
Cross-listed as PMATH 453
AM 433 W 3C 0.5
Differential Geometry
Some global aspects of surface theory, the Euler-Poincaré characteristic, the global interpretation of gaussian curvature via the Gauss-Bonnet formula. Submanifolds of En, induced riemannian metrics, extrinsic and intrinsic curvatures, Gauss-Codazzi equations. Local Lie groups of transformations on Rn, 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 applications to continuum mechanics and mathematical physics.
Prereq: AM 333/PMATH 365 or consent of instructor
Cross-listed as PMATH 465
AM 441 F 3C 0.5
Numeric Computation for Dynamic Simulation
The interaction of continuous dynamic simulation models, numerical methods, and computing environments. Constant coefficient models for restricted operating ranges, linearization. Time stepping techniques for models based on general systems of ordinary differential equations. Stiffness and nonlinearity. Simulation of computer memory circuits. Wave type linear partial differential equations.
Prereq: CS 370 or 337 and consent of instructor
Cross-listed as CS 476
AM 451 W 3C 0.5
Introduction to Dynamical Systems
A unified view of linear and nonlinear systems of ordinary differential equations in Rn. 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-Bendixson theorem and limit cycles. Applications to physical problems will be a motivating influence.
Prereq: AM 251 and 351, or consent of instructor
AM 453 F 3C 0.5
Partial Differential Equations 2
A thorough discussion of the class of second-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: well-posed 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 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
AM 455 F 3C 0.5
Control Theory
Dynamical systems, controllability and observability. Minimization of functions and functionals. Optimal control. Pontryagin's Maximum Principle.
Prereq: AM/PMATH 332 and AM 351 or equivalent, or consent of instructor
AM 456 W 3C 0.5
Calculus of Variations
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 Carathéodory complete figure. Fields and the Legendre and Weierstrass sufficient conditions.
Prereq: AM 231, or consent of instructor
AM 463 F 3C 0.5
Fluid Mechanics
Incompressible, irrotational flow. Incompressible viscous flow. Introduction to wave motion and geophysical fluid mechanics. Elements of compressible flow.
Prereq: AM 361, or consent of instructor
AM 473 F 3C 0.5
Quantum Mechanics 2
Vector space formalism, Schrödinger and Heisenberg pictures, elements of second quantization. Angular momentum, selection rules, symmetry and conservation laws. Approximation methods: variation principle, perturbation theory and WKB approximation. Identical particles, Pauli principle and simple applications to atomic, molecular, solid state, scattering and nuclear problems.
Prereq: AM 373, or consent of instructor
AM 475 W 3C 0.5
Introduction to General Relativity
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 261 and AM 333/PMATH 365 or consent of instructor
AM 475 (after F97) and AM 375 (before F97) may not both be taken for credit
AM 477 W 3C 0.5
Statistical Mechanics
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
AM 495 3C 0.5
Reading Course