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## Applied Mathematics

S.P. Lipshitz, MC 5196, ext. 3755

Courses not offered in the current academic year are listed at the end of this section.

Note: More detailed course descriptions and course outlines are available in the Applied Mathematics Handbook.

AM 231 F,W,S 3C,1T 0.5
Calculus 4
Vector integral calculus, including line integrals, Green's theorem, the Divergence theorem, and Stokes' theorem, with applications to physical problems. Sequences and series of functions and their applications, including the role of uniform convergence.
Prereq: MATH 237
Antireq: MATH 212, 217, 227P

AM 250 F,W 3C 0.5
Modelling with Ordinary Differential Equations
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 108
Antireq: MATH 218, 228
Not available for credit to students in Applied Mathematics programs.
No student is allowed to take all three of AM 250, 251, and 261 for credit.

AM 251 F,W 3C 0.5
Elementary Differential Equations and Applications
Properties of solutions of first- and second-order scalar differential equations; solution techniques. Physical dimensions; scaling; dimensional homogeneity; dimensionless ratios; the Buckingham Pi Theorem. Systems of first-order differential equations in R2; the matrix exponential and linear flow; stability of equilibrium; qualitative analysis; linearization about equilibrium. Applications are drawn from population dynamics and classical mechanics.
Prereq: MATH 138
Coreq: MATH 235
No student is allowed to take all three of AM 250, 251, and 261 for credit.

AM 261 F 3C,1T 0.5
Newtonian Mechanics
Modeling 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
No student is allowed to take all three of AM 250, 251, and 261 for credit.

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,S 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

AM 375 W 3C 0.5
Special Relativity and Electromagnetic Field Theory
Minkowski space-time and Lorentz transformations. Physical consequences. Optics including Doppler effect and observation. Relativistic mechanics. Collisions. E = mc2. Introduction to electricity and magnetism. Maxwell's equations. Tensorial formulation. Four-vector potential and gauge invariance. Algebraic structure of the electromagnetic field. Solutions of Maxwell's equations. Radiation: wave guides and antennae. Energy-momentum tensor of the electromagnetic field.
Prereq: AM 333 and AM 261, or consent of instructor

AM 431 F 3C 0.5
Measure and Integration
Lebesgue measure and integral for the real line, general measure and integration theory, convergence theorems, Fubini's theorem, absolute continuity, Radon Nikodym theorem, LP-spaces.
Prereq: PMATH 351 or 353
Cross-listed as PMATH 451

AM 432 W 3C 0.5
Functional Analysis
Banach spaces, linear operators, geometry of Hilbert spaces, Hahn-Banach theorem, open mapping theorem, compact operators, applications.
Prereq: AM 431/PMATH 451 or PMATH 353
Cross-listed as PMATH 453

AM 433 F or 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
Numerical Solution of Differential and Integral Equations
Initial-value problems: existence and uniqueness of solutions, one step methods, multistep methods, stability, error analysis. Boundary-value problems: shooting and discretization methods, implementation problems especially for non-linear equations. Integral equations: correspondence to ordinary differential equations, initial-value and boundary-value problems, solution techniques.
Prereq: CS 370 or (374, or 337 and consent of instructor, or CS 372 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- Bendixon 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 2nd 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's 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 W 3C 0.5
Control Theory
Dynamical systems, controllability and observability. Minimization of functions and functionals. Optimal control. Pontryagin's Maximum Principle.
Prereq: Consent of instructor

AM 456 F 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 Caratheodory 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 465 W 3C 0.5
Elasticity
Basic equations of elasticity for homogeneous 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 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
Equivalence principle. Curved space-time and Einstein's gravitational field equations. The weak field limit. The Schwarzschild solution. Observational tests of General Relativity. Introduction to relativistic cosmology. Friedmann-Robertson-Walker universes and the big bang. Observational evidence. Gravitational collapse and black holes. Gravitational waves.
Prereq: AM 375, or consent of instructor

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 F 3C 0.5