AMATH 231 Calculus 4
Vector integral calculus-line integrals, surface integrals and vector fields, Green's theorem, the Divergence theorem, and Stokes' theorem. Applications include conservation laws, fluid flow and electromagnetic fields. An introduction to Fourier analysis. Fourier series and the Fourier transform. Parseval's formula. Frequency analysis of signals. Discrete and continuous spectra.
AMATH 250 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.
AMATH 261 Classical Mechanics and Special Relativity
Newtonian dynamics of particles and systems of particles. Oscillations. Gravity and the central force problem. Lorentz transformations and relativistic dynamics.
AMATH 331 Applied Real Analysis
Topology of Euclidean spaces, continuity, norms, completeness. Contraction mapping principle. Fourier series. Various applications, for example, to ordinary differential equations, optimization and numerical approximation.
AMATH 332 Applied Complex Analysis
Complex numbers, Cauchy-Riemann equations, analytic functions, conformal maps and applications to the solution of Laplace's equation, contour integrals, Cauchy integral formula, Taylor and Laurent expansions, residue calculus and applications.
AMATH 333 Elementary Differential Geometry
An introduction to local differential geometry, laying the groundwork
AMATH 341 Introduction to Computational Mathematics
A rigorous introduction to the field of computational mathematics. The focus is on the interplay between continuous models and their solution via discrete processes. Topics include: pitfalls in computation, solution of linear systems, interpolation, discrete Fourier transforms and numerical integration. Applications are used as motivation.
AMATH 342 Computational Methods for Differential Equations
Modelling of systems which lead to differential equations (examples include vibrations, population dynamics, and mixing processes). Scalar first order differential equations, second-order differential equations, systems of differential equations. Stability and qualitative analysis. Implicit and explicit time-stepping. Comparison of different methods. Stiffness. Linearization and the role of the Jacobian.
AMATH 350 Differential Equations for Business and Economics
First order linear and separable differential equations. Exponential growth with applications to continuous compounding. The logistic equation and variations. Introduction to systems of linear differential equations in R2. Dimensional analysis. Linear partial differential equations. Boundary value problems. The diffusion equation. Solutions to the Black-Scholes partial differential equations. Introduction to numerical methods.
AMATH 351 Ordinary Differential Equations 2
Second order linear differential equations with non-constant coefficients, Sturm comparison, oscillation and separation theorems, series solutions and special functions. Linear vector differential equations in Rn, an introduction to dynamical systems. Laplace transforms applied to linear vector differential equations, transfer functions, the convolution theorem. Perturbation methods for differential equations. Numerical methods for differential equations. Applications are discussed throughout.
AMATH 353 Partial Differential Equations 1
Second order linear partial differential equations - the diffusion equation, wave equation, and Laplace's equation. Methods of solution - separation of variables and eigenfunction expansions, the Fourier transform. Physical interpretation of solutions in terms of diffusion, waves and steady states. First order non-linear partial differential equations and the method of characteristics. Applications are emphasized throughout.
AMATH 361 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.
AMATH 373 Quantum Theory 1
Critical experiments and old quantum theory. Basic concepts of quantum mechanics: observables, wavefunctions, Hamiltonians and the Schroedinger equation. Uncertainty, correspondence and superposition principles. Simple applications to finite and extended one-dimensional systems, harmonic oscillator, rigid rotor and hydrogen atom.
AMATH 382 Computational Modeling of Cellular Systems
An introduction to dynamic mathematical modeling of cellular processes. The emphasis is on using computational tools to investigate differential equation-based models. A variety of cellular phenomena are discussed, including ion pumps, membrane potentials, intercellular communication, genetic networks, regulation of metabolic pathways, and signal transduction.
AMATH 391 From Fourier to Wavelets
An introduction to contemporary mathematical concepts in signal analysis. Fourier series and Fourier transforms (FFT), the classical sampling theorem and the time-frequency uncertainty principle. Wavelets and multiresolution analysis. Applications include oversampling, denoising of audio, data compression and singularity detection.
AMATH 431 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.
AMATH 432 Functional Analysis
Banach and Hilbert spaces, bounded linear maps, Hahn-Banach theorem, open mapping theorem, closed graph theorem, topologies, nets, Hausdorff spaces, Tietze extension theorem, dual spaces, weak topologies, Tychonoff's theorem, Banach-Alaoglu theorem, reflexive spaces.
AMATH 433 Differential Geometry
An introduction to differentiable manifolds. The tangent and cotangent bundles. Vector fields and differential forms. The Lie bracket and Lie derivative of vector fields. Exterior differentiation, integration of differential forms, and Stokes's Theorem. Riemannian manifolds, affine connections, and the Riemann curvature tensor.
AMATH 442 Computational Methods for Partial Differential Equations
This course studies basic methods for the numerical solution of partial differential equations. Emphasis is placed on regarding the discretized equations as discrete models of the system being studied. Basic discretization methods on structured and unstructured grids. Boundary conditions. Implicit/explicit timestepping. Stability, consistency and convergence. Non-conservative versus conservative systems. Nonlinearities.
AMATH 447 Introduction to Symbolic Computation
An introduction to the use of computers for symbolic mathematical computation, involving traditional mathematical computations such as solving linear equations (exactly), analytic differentiation and integration of functions, and analytic solution of differential equations.
AMATH 451 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, Poincare-Bendixson theorem and limit cycles. Applications to physical problems will be a motivating influence.
AMATH 453 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.
AMATH 455 Control Theory
Feedback control with applications. System theory in both time and frequency domain, state-space computations, stability, system uncertainty, loopshaping, linear quadratic regulators and estimation.
AMATH 456 Calculus of Variations
Concept of functional and its variations. The solution of problems using variational methods - the Euler-Lagrange equations. Applications include an introduction to Hamilton's Principle and optimal control.
AMATH 463 Fluid Mechanics
Incompressible, irrotational flow. Incompressible viscous flow. Introduction to wave motion and geophysical fluid mechanics. Elements of compressible flow.
AMATH 473 Quantum Theory 2
The Hilbert space of states, observables and time evolution. Feynman path integral and Greens functions. Approximation methods. Coordinate transformations, angular momentum and spin. The relation between symmetries and conservation laws. Density matrix, Ehrenfest theorem and decoherence. Multiparticle quantum mechanics. Bell inequality and basics of quantum computing.
AMATH 475 Introduction to General Relativity
Tensor analysis. Curved space-time and the Einstein field equations. The Schwarzschild solution and applications. The Friedmann-Robertson-Walker cosmological models.