[How to read the course descriptions]

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Undergraduate Officer
*

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

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

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

**Reading Course
**

**AM 496 W 3C 0.5**

**Reading Course
**

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