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2D and 3D geometry, vectors and matrices, eigenvalues and vibration, physical applications. Laboratories demonstrate computer solutions of large systems.
Linear ordinary differential equations, Laplace transforms, Fourier series and separation of variables for linear partial differential equations. Tutorial session focuses on examples from chemical and biological engineering. Equivalency: MECH 221.
Applications of linear algebra to problems in science and engineering; use of computer algebra systems for solving problems in linear algebra.
Parametrizations, inverse and implicit functions, integrals with respect to length and area; grad, div, and curl, theorems of Green, Gauss, and Stokes.
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Numbers and their properties; exponents, radicals, absolute value, inqualities, functions and their graphs; factoring; solving polynomial, rational, and exponetial equations; and the sine and cosine law.
Composite, inverse, polynomial, rational, trigonometric, exponential, and logarithmic functions; sequences and series; and analytical geometry.
Derivatives of elementary functions. Applications and modeling: graphing, optimization.
The definite integral, integration techniques, applications, modeling, infinite series.
Functions, derivatives, optimization, growth and decay, discrete probability.
Antiderivatives and definite integrals, infinite series, applications to probability and dynamical systems.
Derivatives and rates of change, exponential and trigonometric functions, Newton's method, Taylor polynomials, maxima and minima, and graphing.
Antiderivatives, the definite integral, techniques of integration, infinite series, partial derivatives, maxima and minima with constraints, discrete and continuous random variables.
Topics as for MATH 100, but including relevant topics from algebra, geometry, functions, trigonometry, logarithms, and exponentials.
Limits, derivatives, Mean Value Theorem and applications, elementary functions, optimization, Taylor series, approximation.
Definite integrals and the Fundamental Theorem of Calculus, techniques and applications of integration, infinite series.
Topics as for Math 100; intended for students with no previous knowledge of Calculus.
Functions, derivatives, integrals, curve sketching growth functions, volume calculations.
Analytic geometry in 2 and 3 dimensions, partial and directional derivatives, chain rule, maxima and minima, second derivative test, Lagrange multipliers, multiple integrals with applications.
Introduction to numerical computation, computer algebra, mathematical graphics. Primarily for second year students taking a degree in mathematics. One hour laboratory each week.
First-order equations; linear equations; linear systems; Laplace transforms; numerical methods; trajectory analysis of plane nonlinear systems. Applications of these topics will be emphasized.
Partial differentiation, extreme values, multiple integration, vector fields, line and surface integrals, the divergence theorem, Green's and Stokes' theorems.
Sets and functions; induction; cardinality; properties of the real numbers; sequences, series, and limits. Logic, structure, style, and clarity of proofs emphasized throughout.
Systems of linear equations, operations on matrices, determinants, eigenvalues and eigenvectors, diagonalization of symmetric matrices.
Matrices, eigenvectors, diagonalization, orthogonality, linear systems, applications.
Functions of several variables: limits, continuity, differentiability; implicit functions; Taylor's theorem; extrema; Lagrange multipliers; multiple integration, Fubini's theorem; improper integrals.
Parametrization of curves and surfaces; line and surface integrals; theorems of Green, Gauss, Stokes; applications to physics and/or introduction to differential forms.
Difference equations, number theory, counting.
Partial and directional derivatives; maxima and minima; Lagrange multipliers and second derivative test; multiple integrals and applications. Equivalency MECH 222.
Review of linear systems; nonlinear equations and applications; phase plane analysis; Laplace transforms; numerical methods.
Introduction to partial differential equations; Fourier series; the heat, wave and potential equations; boundary-value problems; numerical methods.
Divergence, gradient, curl, theorems of Gauss and Stokes. Applications to Electrostatics and Magnetostatics.
Linear ordinary differential equations. Complex numbers, Laplace transforms, frequency reponse, resonance, step response, systems.
Fourier series and transforms, wave equation, d'Alembert's solution, modes. Discrete Fourier tranform. Recurrence relations, z-transform, generating functions, applications.
Functions of a complex variable, Cauchy-Riemann equations, elementary functions, Cauchy's theorem and contour integration, Laurent series, poles and residues.
Integrals involving multi-valued functions, conformal mapping and applications, analytic continuation, Laplace and Fourier transforms.
Basic notions of probability, random variables, expectation and conditional expectation, limit theorems. Equivalency: UBC STAT 302
Discrete-time Markov chains, Poisson processes, continuous time Markov chains, renewal theory.
Functions of a complex variable, Cauchy-Riemann equations, contour integration, Laurent series, residues, integrals of multi-valued functions, Fourier transforms.
Classical plane geometry, solid geometry, spherical trigonometry, polyhedra, linear and affine transformations. Linear algebra proofs are used.
Topics chosen by the instructor. These may include conic sections, projective configuration, convexity, non-Euclidean geometries, fractal geometry, combinatorial problems of points in the plane.
Linear spaces, duality, linear mappings, matrices, determinant and trace, spectral theory, Euclidean structure.
Euclidean algorithm, congruences, Fermat's theorem, applications. Some diophantine equations. Distribution of the prime numbers.
Topics chosen by the instructor. These might include: division algorithms, group theory, continued fractions, primality testing, factoring.
Power series methods (ordinary and regular singular points, Bessel's equation); boundary value problems and separation of variables (Fourier series and other orthogonal series), applications to the vibrating string, heat flow, potentials.
Random variables, discrete and continuous distributions. Random walk, Markov chains, Monte Carlo methods. Characteristic functions, limit laws.
The real number system; real Euclidean n-space; open, closed, compact, and connected sets; Bolzano-Weierstrass theorem; sequences and series. Continuity and uniform continuity. Differentiability and mean-value theorems.
The Riemann or Riemann-Stieltjes integrals. Sequences and series of functions, uniform convergence. Approximation of continuous functions by polynomials. Fourier series. Functions from Rm to Rn, inverse and implicit function theorems.
Groups, cosets, homomorphisms, group actions, p-groups, Sylow theorems, composition series, finitely generated Abelian groups.
Rings, ideals, unique factorization, Euclidean rings, fields, polynomial rings, modules; structure theory of modules over a principal ideal domain.
Intensive course with required tutorial. Combinatorics, probability, geometry and elementary number theory.
Linear programming problems, dual problems, the simplex algorithm, solution of primal and dual problems, sensitivity analysis. Additional topics chosen from: Karmarkar's algorithm, non-linear programming, game theory, applications.
Introduction to ideas and methods of discrete mathematics and their application.
Error-correcting codes via abstract and linear algebra. Emphasis on proofs and computation. Finite fields, Hamming distance and error-correction, upper and lower bounds on the size of a code, linear codes, groups and cosets, encoding and decoding schemes.
Introduction to mathematical game theory and its applications.
Phase plane methods, bifurcation and stability theory, limit-cycle behavior and chaos for nonlinear differential equations with applications to the sciences. Assignments involve the use of computers.
Fourier series; auto- and cross-correlation; power spectra; discrete Fourier transform; boundary-value problems; numerical methods; partial differential equations; heat, wave, Laplace, Poisson, and wave equations. Applications to mechanical engineering and practical computing applications emphasized.
Principles of model selection and basic modeling techniques in biology, earth science, chemistry and physics. Optimization, dynamical systems and stochastic processes. Preference will be given to Combined Major in Science students, or to students in Year 3 or higher.
Mathematical modeling of basic biological processes in ecology, physiology, neuroscience and genetics. Dynamic behavior of difference equations, differential equations, and partial differential equations, explained with reference to concrete biological examples.
Separation of variables, first order equations, Sturm-Liouville theory, integral transform methods.
Green's functions for partial differential equations. Calculus of variations. Eigenfunction expansions. Rayleigh-Ritz and finite element methods.
Classical variational problems; necessary conditions of Euler, Weierstrass, Legendre, and Jacobi; Erdmann corner conditions, transversality, convex Lagrangians, fields of extremals, sufficient conditions for optimality, numerical methods; applications to classical mechanics, engineering and economics.
Dynamical systems; stability by Liapunov's direct method; controllability and eigenvalue assignment for autonomous linear systems; linear-quadratic regulator, time optimal control, Pontryagin maximum principle, dynamic programming; applications in engineering, economics and resource management.
Interpolation, numerical integration, numerical solution of ordinary and partial differential equations. Practical computational methods emphasized and basic theory developed through simple models.
Variational and Green's function methods for ordinary and partial differential equations, introduction to finite difference, finite element and boundary element methods.
Topics include decompositions of linear operators, multi linear algebra, bilinear forms, metric spaces.
Students will prepare material illustrating ideas and applications of mathematics and present it to audiences outside the University.
Probability spaces, random variables, distributions, expectation, conditional probabilities, convergence of random variables, generating and characteristic functions, weak and strong laws of large numbers, central limit theorem.
Random walks, Markov chains, branching processes, Poisson processes, continuous time Markov chains, martingales, Brownian motion.
Sigma-algebras, Lebesgue measure, Borel measures, measurable functions, integration, convergence theorems, Lp spaces, Holder and Minkowski inequalities, Lebesgue and/or Radon-Nikodym differentiation.
Banach spaces, linear operators, bounded and compact operators, strong, weak, and weak* topology. Hahn-Banach, open mapping, and closed graph theorems. Hilbert spaces, symmetric and self-adjoint operators, spectral theory for bounded operators.
Field extensions, the Galois correspondence, finite fields, insolvability in radicals, ruler and compass constructions, additional topics chosen by instructor.
Commutative algebra, algebraic geometry, algebraic number theory, Lie theory, homological algebra and category theory, or some other advanced topic in algebra.
The differential geometry of curves and surfaces in three-dimensional Euclidean space. Mean curvature and Gaussian curvature. Geodesics. Gauss's Theorema Egregium.
Riemannian manifolds, tensors and differential forms, curvature and geodesics.
General topology, combinatorial topology, fundamental group and covering spaces, topics chosen by the instructor.
Homology theory, homotopy theory, manifolds, and other topics chosen by the instructor.
Newton's equation, conservation laws, the Euler-Lagrange equation; Hamilton's principle of least action, Hamilton's equations, Lagrangian mechanics on manifolds.
The student should consult the Mathematics Department for the particular topics offered in a given year.
The student should consult the Mathematics Department for the particular topics offered in a given year.
Divisibility, congruencies, Diophantine equations, arithmetic functions, quadratic reciprocity, advanced topics.
The residue theorem, the argument principle, conformal mapping, the maximum modulus principle, harmonic functions, representation of functions by integrals, series, and products. Other topics at the discretion of the instructor.
Formulation of real-world optimization problems using techniques such as linear programming, network flows, integer programming, dynamic programming. Solution by appropriate software.
Basic graph theory, emphasizing trees, tree growing algorithms, and proof techniques. Problems chosen from: shortest paths, maximum flows, minimum cost flows, matchings, graph colouring. Linear programming duality will be an important tool.
Introductory course in mostly non-algorithmic topics including: planarity and Kuratowski's theorem, graph colouring, graph minors, random graphs, cycles in graphs, Ramsey theory, extremal graph theory. Proofs emphasized. Intended for Honours students.
Current research topics in pure and applied mathematics are explored at the undergraduate level. Technical communication and research skills are developed.
Historical development of concepts and techniques in areas chosen from Geometry, Number Theory, Algebra, Calculus, Probability, Analysis. The focus is on historically significant writings of important contributors and on famous problems of Mathematics.
Asymptotic expansions. Asymptotic evaluation of integrals; WKBJ methods. Regular and singular expansions. Boundary layer theory; matched asymptotic expansions. Multiple scale techniques.
Development and analysis of mathematical models for complex systems in ecology, evolution, cell biology, neurophysiology, and other biological and medical disciplines.