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, predator-prey dynamics, age-structured populations.
Antirequisite(s): The former Calculus 1201A/B.
Prerequisite(s): One or more of Calculus 1000A/B, 1100A/B, 1500A/B or Mathematics 1225A/B.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 1 tutorial hour, 0.5 course.
 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.
Antirequisite(s): Mathematics 1600A/B
Prerequisite(s): Ontario Secondary School MHF4U or MCV4U, or Mathematics 0110A/B.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 2 computer lab or tutorial hours, 0.5 course.
Restricted to students in the Faculty of Engineering.
 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.
Antirequisite(s): Calculus 1000A/B, 1100A/B, 1301A/B, 1500A/B, 1501A/B, Mathematics 1225A/B.
Prerequisite(s): One or more of Ontario Secondary School MHF4U, MCV4U, or Mathematics 0110A/B.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 1 tutorial hour, 1.0 course.
Applied Mathematics 1413 is a suitable prerequisite for any course which lists Calculus 1000A/B plus Calculus 1501A/B.
Restricted to students in the Faculty of Engineering.
 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.
Antirequisite(s):
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Extra Information: 2 lecture hours, 2 computer lab hours, 0.5 course.
 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.
Prerequisite(s): A minimum mark of 60% in Calculus 1301A/B, or a minimum mark of 55% in Calculus 1501A/B or Applied Mathematics 1413.
Corequisite(s):
Pre-or Corequisite(s): Mathematics 1600A/B or the former Linear Algebra 1600A/B.
Extra Information: 3 lecture hours, 1 laboratory hour, 0.5 course.
 Applied Mathematics 2411 - Applied Mathematics for Engineering II
This course is intended to be taken by Chemical and Civil Engineering students. Topics include ordinary differential equations, Laplace transforms, multiple integrals, introduction to partial differential equations, and Fourier Series.
Antirequisite(s):
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Extra Information: 3 lecture hours, 1 tutorial hour, 1.0 course.
 Applied Mathematics 2413 - Applied Mathematical and Numerical Methods for Mechanical Engineering
Topics include: Introduction to Matlab; numerical differentiation and integration; numerical linear algebra; ordinary differential equations including higher order systems and numerical solutions; interpolation and approximation; multiple integrals and vector integral theorems.
Antirequisite(s): Applied Mathematics 2411, 2415, 2813B.
Prerequisite(s):
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Extra Information: 3 lecture hours, 1.5 laboratory hours, 1.0 course.
 Applied Mathematics 2415 - Applied Mathematical Methods for Electrical and Software Engineering I
Topics include: ordinary differential equations methods including Laplace transforms; Fourier series and transforms; multiple integration; vector fields, line integrals; vector calculus including Green's and Stokes's theorems; computer applications.
Antirequisite(s):
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Extra Information: 3 lecture hours, 1.5 laboratory hours, 1.0 course.
 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 predator-prey, business competition, coupled oscillators. Singular value decomposition, image approximations. Linear transformations, graphics.
Antirequisite(s):
Prerequisite(s): Applied Mathematics 1413 or Calculus 1301A/B or 1501A/B and a minimum mark of 60% in Mathematics 1600A/B or the former Linear Algebra 1600A/B, or Applied Mathematics 1411A/B.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 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.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 1 laboratory hour, 0.5 course.
 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.
Antirequisite(s):
Prerequisite(s): Mathematics 1600A/B or the former Linear Algebra 1600A/B or equivalent, Calculus 2303A/B or 2503A/B.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 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, Hamilton-Jacobi theory, constrained systems, field theory.
Antirequisite(s):
Prerequisite(s): Calculus 2503A/B, Mathematics 1600A/B or the former Linear Algebra 1600A/B, and one of either Physics 1301A/B and 1302A/B, or Physics 1401A/B and 1402A/B, or Physics 1501A/B and Physics 1502A/B, or the former Physics 1020, 1024 or 1026.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 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.
Antirequisite(s):
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Extra Information: 3 lecture hours, 0.5 course.
 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.
Antirequisite(s):
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Extra Information: 3 lecture hours, 1 laboratory hour, 0.5 course.
 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.
Antirequisite(s):
Prerequisite(s): One of Calculus 2302A/B, 2402A/B, 2502A/B; plus one of Mathematics 1600A/B or the former Linear Algebra 1600A/B, or Applied Mathematics 1411A/B.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 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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Extra Information: 3 lecture hours, 0.5 course.
 Applied Mathematics 3813A/B - Nonlinear Ordinary Differential Equations and Chaos
Existence and uniqueness of solutions, phase space, singular points, stability, periodic attractors, Poincaré-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.
Antirequisite(s):
Prerequisite(s): Applied Mathematics 2402A or the former Differential Equations 2402A; Calculus 2303A/B or 2503A/B and Mathematics 1600A/B or the former Linear Algebra 1600A/B.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 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; Sturm-Liouville Theory; eigenfunction expansions; cylindrical and spherical problems; Legendre and Bessel functions; spherical harmonics; Fourier and Laplace transforms.
Antirequisite(s):
Prerequisite(s): (i) Mathematics 1600A/B or the former Linear Algebra 1600A/B; Applied Mathematics 2402A or the former Differential Equations 2402A; Calculus 2303A/B or 2503A/B;  or (ii) Calculus 2402A/B and Applied Mathematics 2503A/B.  In each course a minimum mark of 60% is required.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 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.
Antirequisite(s):
Prerequisite(s):
Corequisite(s): Calculus 2303A/B or 2503A/B, or equivalent, and Applied Mathematics 2814F/G or the former 2813B or Statistical Sciences 2864A/B.
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 1 laboratory hour, 0.5 course.
 Applied Mathematics 4129A/B - Fluid Dynamics
An introduction to ideal and viscous incompressible flows. Some exact and self-similar solutions of Navier Stokes equations, Boundary layer theory and Blasius solution.
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Extra Information: 3 lecture hours, 0.5 course.
 Applied Mathematics 4151A/B - Advanced Classical Mechanics II
Hamilton's equations, canonical transformations, symplectic space, Poisson brackets, integrability, Liouville's theorem, Hamilton-Jacobi theory, chaos, classical field theory.
Antirequisite(s):
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Extra Information: 3 lecture hours, 0.5 course.
May be offered in alternate years.
 Applied Mathematics 4251A - Quantum Mechanics II
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.
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Extra Information: 3 lecture hours, 0.5 course.
May be offered in alternate years.
 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 fermi-gas applications such as atomic nuclei, white dwarfs, and neutron stars; further applications of quantum mechanics.
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Extra Information: 3 lecture hours, 0.5 course.
May be offered in alternate years.
 Applied Mathematics 4351A - Electromagnetic Theory II
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.
Antirequisite(s):
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Extra Information: 3 lecture hours, 0.5 course.
May be offered in alternate years.
 Applied Mathematics 4353B - Classical Field Theory
Hamilton's Principle; Lagrangian for continuous systems; relativistic theories of particles and fields, Green's functions; Lienard-Wiechert 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 co-ordinates; introduction to general relativity.
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Extra Information: 3 lecture hours, 0.5 course.
May be offered in alternate years.
 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.
Antirequisite(s):
Prerequisite(s): Permission of the Department.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
May be offered in alternate years.
 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.
Antirequisite(s):
Prerequisite(s): Physics 2101A/B and 2102A/B, or Physics 2128A/B and 2129A/B; Calculus 2503A/B.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
Typically offered in alternate years only.
 Applied Mathematics 4611F/G - Introduction to Object Oriented Scientific Programming
Basic introduction to C++, review of numerical methods applicable to problems in linear algebra and differential equations, introduction to the concept of object-oriented programming techniques, applications to scientific computation. Grade is based upon two projects and a presentation.
Antirequisite(s):
Prerequisite(s): Calculus 1301A/B, 1501A/B, or Applied Mathematics 1413; and Applied Mathematics 2413, 2415 or 2814F/G or the former 2813B.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
Offered in alternate years with Applied Mathematics 4615F/G.
 Applied Mathematics 4613A/B - Finite Element Methods
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.
Antirequisite(s):
Prerequisite(s): Applied Mathematics 2814F/G or the former 2813B.
Corequisite(s): Applied Mathematics 3815A/B or equivalent.
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
Offered in alternate years with Applied Mathematics 4617A/B.
 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.
Antirequisite(s):
Prerequisite(s): Applied Mathematics 2413, 2415 or 2814F/G or the former 2813B.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
Offered in alternate years with Applied Mathematics 4611F/G.
 Applied Mathematics 4617A/B - Numerical Solutions of Partial Differential Equations
Finite difference methods, stability analysis for time-dependent problems.
Antirequisite(s):
Prerequisite(s): Applied Mathematics 2413 or 2814F/G or the former 2813B.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
Offered in alternate years with Applied Mathematics 4613A/B.
 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.
Antirequisite(s):
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Extra Information: 3 lecture hours, 0.5 course.
May be offered in alternate years.
 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.
Antirequisite(s):
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Extra Information: 3 lecture hours, 0.5 course.
May be offered in alternate years.
 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, self-adjoint, unitary, compact and projection operators, spectral decomposition of self-adjoint operators; Fredholm integral equations; mathematical foundations of Quantum Mechanics.
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Extra Information: 3 lecture hours, 0.5 course.