AMATH Course Descriptions

Applied Mathematics (2009-2010)

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. [Offered: F,W,S]

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. [Offered: F,W,S]

Newtonian dynamics of particles and systems of particles. Oscillations. Gravity and the central force problem. Lorentz transformations and relativistic dynamics. [Offered: W,S]

Topology of Euclidean spaces, continuity, norms, completeness. Contraction mapping principle. Fourier series. Various applications, for example, to ordinary differential equations, optimization and numerical approximation.

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.

An introduction to local differential geometry, laying the groundwork for both global differential geometry and general relativity. Submanifolds of n-dimensional Euclidean space. Embedded curves and the intrinsic geometry of surfaces in Euclidean 3-space. Metrics, geodesics, and curvature. Gaussian curvature and the Gauss-Bonnet theorem. [Offered: W]

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.

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. [Offered: F,S]

Difference equations, Laplace and z transforms applied to discrete (and continuous) mathematical models taken from ecology, biology, economics and other fields. [Offered: W]

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. [Offered: F]

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. [Offered: F,S]

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. [Offered: W,S]

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. [Offered: W]

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. [Offered: W]

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.

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.

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. [Offered: W]

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. [Offered: F]

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.

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. [Offered: F]

This course will present two major applications of differential equations based modeling, and focus on the specific problems encountered in each application area. The areas may vary from year to year. Students will gain some understanding of the steps involved in carrying out a realistic numerical modelling exercise. Possible areas include: Fluid Dynamics, Finance, Control, Acoustics, Fate and Transport of Environmental Contaminants. [Offered: W]

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.

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. [Offered: W]

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.

Feedback control with applications. System theory in both time and frequency domain, state-space computations, stability, system uncertainty, loopshaping, linear quadratic regulators and estimation. [Offered: W]

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. [Offered: F]

Incompressible, irrotational flow. Incompressible viscous flow. Introduction to wave motion and geophysical fluid mechanics. Elements of compressible flow. [Offered: F]

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. [Offered: F]

Tensor analysis. Curved space-time and the Einstein field equations. The Schwarzschild solution and applications. The Friedmann-Robertson-Walker cosmological models. [Offered: W]

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. [Offered: W]