Applications of integration, integration using mathematical software packages. Scaling and allometry. Basic probability theory. Fundamentals of linear algebra: vectors, matrices, matrix algebra. Difference and differential equations. Each topic will be illustrated by examples and applications from the biological sciences, such as population growth, predator-prey dynamics, age-structured populations.

Matrix operations, systems of linear equations, linear spaces and transformations, determinants, eigenvalues and eigenvectors, applications of interest to Engineers including diagonalization of matrices, quadratic forms, orthogonal transformations; introduction to MATLAB with applications from linear algebra.

Limits, continuity, differentiation of functions of one variable with applications, extreme values, integration, the fundamental theorem of calculus, methods and applications of integration to areas, volumes and engineering applications. Sequences and series, convergence, power series. Vector functions, partial differential calculus, gradients, directional derivatives and applications.

Behind the polished presentations of most mathematical results there often lie dramatically powerful experimental methods. Modern computational tools have vastly increased the effectiveness of this approach. This course provides tools and opportunities for experiment and the discovery of new mathematics. The best projects from this course will be published.

Topics include first order ODE's of various types, higher order ODE's and methods of solving them, initial and boundary value problems, applications to mass-spring systems and electrical RLC circuits, Laplace transforms and their use for solving differential equations, systems of linear ODE's, orthogonal functions and Fourier.

Topics covered include a review of orthogonal expansions of functions and Fourier series and transforms, multiple integration with methods of evaluation in different systems of coordinates, vector fields, line integrals, surface and flux integrals, the Green, Gauss and Stokes theorems with applications.

Topics covered include a review of orthogonal expansions of functions and Fourier series, partial differential equations and Fourier series solutions, boundary value problems, the wave, diffusion and Laplace equations, multiple integration with methods of evaluation in different systems of coordinates, vector fields, line integrals, surface and flux integrals, the Green, Gauss and Stokes theorems with applications.

Introduction to first order differential equations, linear second and higher order differential equations with applications, complex numbers including Euler's formula, series solutions, Bessel and Legendre equations, existence and uniqueness, introduction to systems of linear differential equations.

Vector space examples. Inner products, orthogonal sets including Legendre polynomials, trigonometric functions, wavelets. Projections, least squares, normal equations, Fourier approximations. Eigenvalue problems, diagonalization, defective matrices. Coupled difference and differential equations; applications such as predator-prey, business competition, coupled oscillators. Singular value decomposition, image approximations. Linear transformations, graphics.

Introduction to numerical analysis; polynomial interpolation, numerical integration, matrix computations, linear systems, nonlinear equations and optimization, the initial value problem. Assignments using a computer and the
software package, Matlab, are an important component of this course.

This course provides students with the tools to tackle more complex problems than those covered in introductory mechanics. D'Alembert's principle, principle of least action, Lagrange's equations, Hamilton's equations, Poisson brackets, canonical transformations, central forces, rigid bodies, oscillations. Optional topics including: special relativity, Hamilton-Jacobi theory, constrained systems, field theory.

Topics include: introduction to complex analysis; complex integration; Fourier series, integrals and transforms; boundary value problems; separation of variables; transform methods of solution for PDE's; applications to mechanical engineering.

Topics Include: introduction to complex analysis; complex integration; boundary value problems; separation of variables; Fourier series and transform methods of solution for PDE's, applications to electrical engineering.

Basic introduction to C++ and the concept of object-oriented programming techniques. Applications to scientific computation applied to numerical methods, linear algebra and differential equations. Grade is largely based on projects and presentations.

An introduction to mathematical biology. Case studies from neuroscience,immunology, medical imaging, cell biology, molecular evolution and ecology will give an overview of this diverse field, illustrating standard mathematical approaches such as compartmental analysis and evolutionary game theory.

Functions of a complex variable, analytic functions, integration in the complex plane, Taylor and Laurent series, analytic continuation, Cauchy's theorem, evaluation of integrals using residue theory, applications to Laplace transforms, conformal mapping and its applications.

Existence and uniqueness of solutions, phase space, singular points, stability, periodic attractors, PoincarĂ©-Bendixson theorem, examples from physics, biology and engineering, frequency (phase) locking, parametric resonance, Floquet theory, stability of periodic solutions, strange attractors and chaos, Lyapunov exponents, chaos in nature, fractals.

Boundary value problems for Laplace, heat, and wave equations; derivation of equations; separation of variables; Fourier series; Sturm-Liouville Theory; eigenfunction expansions; cylindrical and spherical problems; Legendre and Bessel functions; spherical harmonics; Fourier and Laplace transforms.

Basic principles of modelling and simulation, description and treatment of deterministic and random processes, computational methods and applications with emphasis on the use of computers. The course includes a major project.

Corequisite(s): Calculus 2303A/B or Calculus 2503A/B, or equivalent, and Applied Mathematics 2814F/G or the former Applied Mathematics 2813B or Statistical Sciences 2864A/B.

Quantum mechanical description of angular momentum; Stern-Gehrlach experiment and electron spin; addition of angular momenta; full separation of variables treatment of the hydrogen atom Schrodinger equation; time independent non-degenerate and degenerate perturbation theory; fermions, antisymmetry, and the helium atom; time-dependent perturbation theory, Fermi golden rule, and radiative transitions.

Static fields (Green's functions); time varying fields; Maxwell's equations, conservation laws; non-relativistic motion of particle in static, uniform external fields; Rutherford scattering; plane waves; simple radiating systems; fields of a moving charge; relativistic formulation.

Phenomenology; conservation laws and invariance principles; analysis of reactions and decays; the identification of particles; the particle spectrum; unitary symmetry; quarks; models of strong interaction dynamics.

Variational principles, methods of approximation, basis functions, convergence of approximations, solution of steady state problems, solution of time-dependent problems. Each student will be required to complete two major computational projects.

Strengths and limitations of computer algebra systems (CAS); complexity of exact computations versus possible instability of numerical computations; selecta from Groebner bases, resultants, fractional derivatives, Risch integration algorithm, special functions including the Lambert W function. The emphasis is on preparing the student to use CAS in mathematics, science, and engineering.

Finite difference methods, stability analysis for time-dependent problems.

Boundary value problems for Laplace and Helmholtz equations, initial value problems for heat and wave equations, in one to three dimensions; Green's functions in bounded and unbounded domains; Method of Images.

Fourier, Laplace and Hankel transforms with applications to partial differential equations; integral equations; and signal processing and imaging; asymptotic methods with application to integrals and differential equations.

The student will work on a project under faculty supervision. The project may involve an extension, or more detailed coverage, of material presented in other courses. Credit for the course will involve a written as well as oral presentation.