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Applied Mathematics
Calculus 050a/b, 051a/b, 081a/b, 250a/b, 251a/b, 280a/b, and 281a/b are offered jointly by the Departments of Applied Mathematics and Mathematics.
Calculus 091a/b is offered by the Department of Applied Mathematics.
Differential Equations 215a is offered jointly by the Departments of Applied Mathematics and Mathematics.
Linear Algebra 040a/b is offered jointly by the Departments of Applied Mathematics and Mathematics.
Please refer to the CALCULUS (S), DIFFERENTIAL EQUATIONS (S), and LINEAR ALGEBRA (S) subjects for those course offerings. |
| Applied Mathematics 025a/b, Linear Algebra for Engineers | |
| | Description: 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. For students in Engineering Science only. | | Prerequisite(s): Grade 12U Geometry and Discrete Mathematics (MGA4U), or Mathematics 017a/b; Grade 12U Advanced Functions and Introductory Calculus (MCB4U), or Mathematics 012a/b. | | 3 lecture hours, 1 tutorial hour, 0.5 course. | | back to top |
| Applied Mathematics 213b, Linear Algebra II | |
| | Description: 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. | | Prerequisite(s): Applied Mathematics 026 or Calculus 051a/b or 081a/b and a minimum mark of 60% in Linear Algebra 040a/b or Applied Mathematics 025a/b. | | 3 lecture hours, 0.5 course. | | back to top |
| Applied Mathematics 261b, Numerical Analysis | |
| | Description: 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 275. | | Prerequisite(s): A minimum mark of 55% in Linear Algebra 040a/b. | | Corequisite(s): Calculus 250a/b or 280a/b. | | It is recommended that Differential Equations 215a be taken prior to this course. | | 3 lecture hours, 1 laboratory hour, 0.5 course. | | back to top |
| Applied Mathematics 275, Applied Mathematical and Numerical Methods for Mechanical Engineering | |
| | Description: 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 261b, 276, 277. | | Prerequisite(s): Applied Mathematics 025a/b and 026. | | 3 lecture hours, 1.5 laboratory hours, 1.0 course. | | back to top |
| Applied Mathematics 276, Applied Mathematical Methods for Electrical and Software Engineering I | |
| | Description: 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): Applied Mathematics 275, 277. | | Prerequisite(s): Applied Mathematics 025a/b and 026. | | 3 lecture hours, 1.5 laboratory hours, 1.0 course. | | back to top |
| Applied Mathematics 277, Applied Mathematics for Engineering II | |
| | Description: 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): Applied Mathematics 275, 276, Calculus 250a/b plus 251a/b, Calculus 280a/b plus 281a/b, Applied Mathematics 290a plus 291b. | | Prerequisite(s): Applied Mathematics 025a/b and 026. | | 3 lecture hours, 1.0 course. | | back to top |
| Applied Mathematics 301a/b, Complex Variables with Applications | |
| | Description: 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. | | Antirequisite(s): Mathematics 307b. | | Prerequisite(s): Calculus 251a/b, Calculus 281a/b, or Applied Mathematics 291b. | | 3 lecture hours, 0.5 course. | | back to top |
| Applied Mathematics 303a/b, Nonlinear Ordinary Differential Equations and Chaos | |
| | Description: 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. | | Prerequisite(s): Differential Equations 215a, Calculus 251a/b or Calculus 281a/b, and Linear Algebra 040a/b. | | 3 lecture hours, 0.5 course. | | back to top |
| Applied Mathematics 310F/G, Modelling and Simulation | |
| | Description: 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): Physics 326F/G. | | Corequisite(s): Calculus 251a/b, 281a/b, or equivalent, and Applied Mathematics 261b or Statistical Sciences 259a/b. | | 3 lecture hours, 1 laboratory hour, 0.5 course. | | back to top |
| Applied Mathematics 311a/b, Mathematical Biology | |
| | Description: 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. | | Prerequisite(s): One of Calculus 250a/b, Calculus 280a/b, or Applied Mathematics 290a; plus one of Linear Algebra 040a/b, Applied Mathematics 025a/b, or Applied Mathematics 291b | | 3 lecture hours, 0.5 course. | | back to top |
| Applied Mathematics 315a/b, Partial Differential Equations I | |
| | Description: 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. | | Prerequisite(s): Minimum mark of 60% in each of the following: Linear Algebra 040a/b, Differential Equations 215a, Calculus 251a/b or 281a/b. | | 3 lecture hours, 0.5 course. | | back to top |
| Applied Mathematics 320b, Mathematics of Financial Options | |
| | Description: An introduction to modern financial mathematics using a differential equations approach. Stocks, bonds, forwards, futures, and options. Geometric Brownian Motion. No-arbitrage options pricing. The Black-Scholes equation and its solutions. American options and moving boundary problems. | | Antirequisite(s): Statistical Sciences 420a/b | | Prerequisite(s): Applied Mathematics 315a/b or Applied Mathematics 375a/b | | 3 lecture hours, 0.5 course. | | back to top |
| Applied Mathematics 325a/b, Optimization | |
| | Description: An introduction to linear programming, simplex method, duality theory and sensitivity analysis, formulating linear programming models, nonlinear optimization, unconstrained and constrained optimization, quadratic programming. Applications. | | Antirequisite(s): Mathematics 236, the former Statistical Sciences 236. | | Prerequisite(s): Calculus 250a/b or 280a/b, Linear Algebra 040a/b. | | 3 lecture hours, 0.5 course. | | May be offered in alternate years. | | back to top |
| Applied Mathematics 351a/b, Introduction to Continuum Mechanics | |
| | Description: Introduction to Continuum Mechanics. The concept of a continuum. Derivation of the fundamental equations describing a continuum. Application to fluids and solids. | | Prerequisite(s): Linear Algebra 040a/b or equivalent, Calculus 251a/b or Calculus 281a/b. | | 3 lecture hours, 0.5 course. | | back to top |
| Applied Mathematics 353a/b, Classical Mechanics I | |
| | Description: 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): Physics 350a/b. | | Prerequisite(s): Calculus 251a/b, Linear Algebra 040a/b, and one of Physics 020, 024, the former Physics 025 or 026. | | 3 lecture hours, 0.5 course. | | back to top |
| Applied Mathematics 356a, Quantum Mechanics II | |
| | Description: 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. | | Antirequisite(s): Physics 451a/b. | | Prerequisite(s): Physics 351a/b. | | 3 lecture hours, 0.5 course. | | May be offered in alternate years. | | back to top |
| Applied Mathematics 375a/b, Applied Mathematics for Mechanical Engineers | |
| | Description: Topics include: Fourier series, integrals and transforms; boundary value problems in cartesian coordinates; separation of variables; Fourier and Laplace methods of solution. | | Antirequisite(s): Applied Mathematics 376a/b. | | Prerequisite(s): Applied Mathematics 275. | | 3 lecture hours, 0.5 course. | | back to top |
| Applied Mathematics 376a/b, Applied Mathematics for Electrical Engineers | |
| | Description: Topics include: Introduction to complex analysis; complex integration; vector calculus including integral theorems; boundary value problems in cartesian coordinates; separation of variables; Fourier series and transform methods of solution. | | Antirequisite(s): Applied Mathematics 375a/b. | | Prerequisite(s): Applied Mathematics 276. | | 3 lecture hours, 1 laboratory hour, 0.5 course. | | back to top |
| Applied Mathematics 402a/b, Linear Operators for Physical Science | |
| | Description: 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. | | Antirequisite(s): Mathematics 418a/b. | | 3 lecture hours, 0.5 course. | | back to top |
| Applied Mathematics 415a/b, Partial Differential Equations II | |
| | Description: 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. | | Prerequisite(s): Applied Mathematics 315a/b. | | 3 lecture hours, 0.5 course. | | May be offered in alternate years. | | back to top |
| Applied Mathematics 420b, Electromagnetic Theory II | |
| | Description: 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): Physics 420a/b. | | Prerequisite(s): Physics 365a/b. | | 3 lecture hours, 0.5 course. | | May be offered in alternate years. | | back to top |
| Applied Mathematics 422b, Classical Field Theory | |
| | Description: 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. | | Prerequisite(s): Physics 365a/b. | | 3 lecture hours, 0.5 course. | | May be offered in alternate years. | | back to top |
| Applied Mathematics 425a/b, Methods of Applied Mathematics | |
| | Description: 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. | | Prerequisite(s): Applied Mathematics 315a/b. | | Pre- or Corequisite(s): Applied Mathematics 301a/b. | | 3 lecture hours, 0.5 course. | | May be offered in alternate years. | | back to top |
| Applied Mathematics 432a/b, Fluid Dynamics | |
| | Description: An introduction to ideal and viscous incompressible flows. Some exact and self-similar solutions of Navier Stokes equations, Boundary layer theory and Blasius solution. | | Prerequisite(s): Applied Mathematics 351a/b. | | 3 lecture hours, 0.5 course. | | back to top |
| Applied Mathematics 453a/b, Advanced Classical Mechanics II | |
| | Description: Hamilton's equations, canonical transformations, symplectic space, Poisson brackets, integrability, Liouville's theorem, Hamilton-Jacobi theory, chaos, classical field theory. | | Prerequisite(s): Applied Mathematics 353a/b or Physics 350a/b. | | 3 lecture hours, 0.5 course. | | May be offered in alternate years. | | back to top |
| Applied Mathematics 455a/b, Introduction to Elementary Particles | |
| | Description: 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): Physics 454a/b. | | Prerequisite(s): Permission of the Department. | | 3 lecture hours, 0.5 course. | | May be offered in alternate years. | | back to top |
| Applied Mathematics 456b, Quantum Mechanics III | |
| | Description: 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. | | Antirequisite(s): The fomer Physics 461a/b. | | Prerequisite(s): Physics 351a/b. | | Corequisite(s): Applied Mathematics 356b or the former Physics 352b. | | 3 lecture hours, 0.5 course. | | May be offered in alternate years. | | back to top |
| Applied Mathematics 465F/G, Introduction to Object Oriented Scientific Programming | |
| | Description: 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): The former Applied Mathematics 465a/b. | | Prerequisite(s): Calculus 051a/b, 081a/b, or Applied Mathematics 026; and Applied Mathematics 261b, 275, or 276. | | 3 lecture hours, 0.5 course. | | Offered in alternate years with Applied Mathematics 475a/b. | | back to top |
| Applied Mathematics 466a/b, Finite Element Methods | |
| | Description: 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. | | Prerequisite(s): Applied Mathematics 261b. | | Corequisite(s): Applied Mathematics 315a/b or equivalent. | | 3 lecture hours, 0.5 course. | | Offered in alternate years with Applied Mathematics 476a/b. | | back to top |
| Applied Mathematics 475F/G, Introduction to Applied Computer Algebra | |
| | Description: 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. | | Prerequisite(s): pplied Mathematics 261b, Applied Mathematics 275, Applied Mathematics 276, or the former Applied Mathematics 272a/b, 372a/b. | | 3 lecture hours, 0.5 course. | | Offered in alternate years with Applied Mathematics 465F/G. | | back to top |
| Applied Mathematics 491Z, Project | |
| | Description: 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): The former Applied Mathematics 491y. | | Prerequisite(s): Registration in the fourth year of a program in Applied Mathematics. | | 0.5 course. | | back to top |
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