 
Applied Mathematics (S)

Please refer to the CALCULUS (S) section for other course offerings from the Department of Applied Mathematics.


Applied Mathematics
1201A/B 
Calculus and Probability with Biological Applications

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, predatorprey dynamics, agestructured populations.
Antirequisite(s):
The former Calculus 1201A/B.
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Applied Mathematics
1411A/B 
Linear Algebra with Numerical Analysis for Engineering

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.
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Applied Mathematics
1413 
Applied Mathematics for Engineers I

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.
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Applied Mathematics
1999F/G 
Introduction to Experimental Mathematics

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.
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Applied Mathematics
2270A/B 
Applied Mathematics for Engineering II

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 massspring systems and electrical RLC circuits, Laplace transforms and their use for solving differential equations, systems of linear ODE's, orthogonal functions and Fourier series.
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Applied Mathematics
2276A/B 
Applied Mathematics for Electrical and Mechanical Engineering III

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.
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Applied Mathematics
2277A/B 
Applied Mathematics for Chemical and Civil Engineering III

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.
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Applied Mathematics
2402A 
Ordinary Differential Equations

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.
Antirequisite(s):
The former Differential Equations 2402A.
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Applied Mathematics
2811B 
Linear Algebra II

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 predatorprey, business competition, coupled oscillators. Singular value decomposition, image approximations. Linear transformations, graphics.
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Applied Mathematics
2814F/G 
Numerical Analysis

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.
Antirequisite(s):
Applied Mathematics 2413, the former Applied Mathematics 2813B.
Prerequisite(s):
A minimum mark of 55% in Mathematics 1600A/B or the former Linear Algebra 1600A/B.
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Applied Mathematics
3129A/B 
Introduction to Continuum Mechanics

Introduction to Continuum Mechanics. The concept of a continuum. Derivation of the fundamental equations describing a continuum. Application to fluids and solids.
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Applied Mathematics
3151A/B 
Classical Mechanics I

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, HamiltonJacobi theory, constrained systems, field theory.
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Applied Mathematics
3413A/B 
Applied Mathematics for Mechanical Engineers

Topics include: Fourier series, integrals and transforms; boundary value problems in cartesian coordinates; separation of variables; Fourier and Laplace methods of solution.
Prerequisite(s):
Applied Mathematics 2413.
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Applied Mathematics
3415A/B 
Applied Mathematics for Electrical Engineering II

Topics Include: numerical methods; introduction to complex analysis; complex integration; boundary value problems in cartesian coordinates; separation of variables; Fourier series and transform methods of solution.
Prerequisite(s):
Applied Mathematics 2415.
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Applied Mathematics
3611F/G 
Introduction to Object Oriented Scientific Programming

Basic introduction to C++ and the concept of objectoriented programming techniques. Applications to scientific computation applied to numerical methods, linear algebra and differential equations. Grade is largely based on projects and presentations.
Antirequisite(s):
The former Applied Mathematics 4611F/G.
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Applied Mathematics
3615A/B 
Mathematical Biology

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.
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Applied Mathematics
3811A/B 
Complex Variables with Applications

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.
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Applied Mathematics
3813A/B 
Nonlinear Ordinary Differential Equations and Chaos

Existence and uniqueness of solutions, phase space, singular points, stability, periodic attractors, PoincaréBendixson 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.
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Applied Mathematics
3815A/B 
Partial Differential Equations I

Boundary value problems for Laplace, heat, and wave equations; derivation of equations; separation of variables; Fourier series; SturmLiouville Theory; eigenfunction expansions; cylindrical and spherical problems; Legendre and Bessel functions; spherical harmonics; Fourier and Laplace transforms.
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Applied Mathematics
3911F/G 
Modelling and Simulation

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.
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Applied Mathematics
4129A/B 
Fluid Dynamics

An introduction to ideal and viscous incompressible flows. Some exact and selfsimilar solutions of Navier Stokes equations, Boundary layer theory and Blasius solution.
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Applied Mathematics
4151A/B 
Advanced Classical Mechanics II

Hamilton's equations, canonical transformations, symplectic space, Poisson brackets, integrability, Liouville's theorem, HamiltonJacobi theory, chaos, classical field theory.
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Applied Mathematics
4251A 
Quantum Mechanics II

Quantum mechanical description of angular momentum; SternGehrlach experiment and electron spin; addition of angular momenta; full separation of variables treatment of the hydrogen atom Schrodinger equation; time independent nondegenerate and degenerate perturbation theory; fermions, antisymmetry, and the helium atom; timedependent perturbation theory, Fermi golden rule, and radiative transitions.
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Applied Mathematics
4253B 
Quantum Mechanics III

Scattering theory, partial wave analysis, and phase shifts; Dirac equation and the magnetic moment of the electron; many particle systems and fermigas applications such as atomic nuclei, white dwarfs, and neutron stars; further applications of quantum mechanics.
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Applied Mathematics
4351A 
Electromagnetic Theory II

Static fields (Green's functions); time varying fields; Maxwell's equations, conservation laws; nonrelativistic motion of particle in static, uniform external fields; Rutherford scattering; plane waves; simple radiating systems; fields of a moving charge; relativistic formulation.
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Applied Mathematics
4353B 
Classical Field Theory

Hamilton's Principle; Lagrangian for continuous systems; relativistic theories of particles and fields, Green's functions; LienardWiechert potential; motion of charges in electromagnetic fields; electromagnetic field tensor; Lorentz transformations of electromagnetic fields; action function of electromagnetic fields; Noether's theorem; gravitational field in relativistic mechanics; curvilinear coordinates; introduction to general relativity.
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Applied Mathematics
4551A/B 
Introduction to Elementary Particles

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.
Prerequisite(s):
Permission of the Department.
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Applied Mathematics
4602A/B 
Gravitational Astrophysics and Cosmology

Introduction to gravity in astrophysics. Application of Newtonian gravitation to basic galactic dynamics and galactic structure. An introduction to general relativity with applications to black holes, cosmology, and the early universe.
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Applied Mathematics
4613A/B 
Finite Element Methods

Variational principles, methods of approximation, basis functions, convergence of approximations, solution of steady state problems, solution of timedependent problems. Each student will be required to complete two major computational projects.
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Applied Mathematics
4615F/G 
Introduction to Applied Computer Algebra

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.
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Prerequisite(s):
Applied Mathematics 2413, 2415 or 2814F/G or the former 2813B.
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Applied Mathematics
4617A/B 
Numerical Solutions of Partial Differential Equations

Finite difference methods, stability analysis for timedependent problems.
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Applied Mathematics 2413 or 2814F/G or the former 2813B.
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Applied Mathematics
4815A/B 
Partial Differential Equations II

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.
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Applied Mathematics
4817A/B 
Methods of Applied Mathematics

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.
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Applied Mathematics
4819A/B 
Linear Operators for Physical Science

Introduction to infinite dimensional linear spaces and their occurrence in applications; metric and Banach spaces: bounded operators; Volterra integral equation; introduction to the Lebesgue integral; Hilbert space, selfadjoint, unitary, compact and projection operators, spectral decomposition of selfadjoint operators; Fredholm integral equations; mathematical foundations of Quantum Mechanics.
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Applied Mathematics
4999Z 
Project

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.
Antirequisite(s):
Prerequisite(s):
Registration in the fourth year of a program in Applied Mathematics.
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