## Mathematics Community home page

A place to share course materials including videos, photos, tutorials, syllabi and other tools to assist with teaching and learning across all areas of undergraduate mathematics.*Please review the license information provided for each item as usage rights vary.*

### Collections in this community

#### UBC-MATH 152: Linear Systems

2D and 3D geometry, vectors and matrices, eigenvalues and vibration, physical applications. Laboratories demonstrate computer solutions of large systems.

#### UBC-MATH 256: Differential equations

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.

#### UBC-MATH 307: Applied Linear Algebra

Applications of linear algebra to problems in science and engineering; use of computer algebra systems for solving problems in linear algebra.

#### UBC-MATH 317: Calculus IV

Parametrizations, inverse and implicit functions, integrals with respect to length and area; grad, div, and curl, theorems of Green, Gauss, and Stokes.

#### Empty Collections (show/hide)

#### UBC-MATH 001: Algebra

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.

#### UBC-MATH 002: Pre-Calculus

Composite, inverse, polynomial, rational, trigonometric, exponential, and logarithmic functions; sequences and series; and analytical geometry.

#### UBC-MATH 100: Differential Calculus with Applications to Physical Sciences and Engineering

Derivatives of elementary functions. Applications and modeling: graphing, optimization.

#### UBC-MATH 101: Integral Calculus with Applications to Physical Sciences and Engineering

The definite integral, integration techniques, applications, modeling, infinite series.

#### UBC-MATH 102: Differential Calculus with Applications to Life Sciences

Functions, derivatives, optimization, growth and decay, discrete probability.

#### UBC-MATH 103: Integral Calculus with Applications to Life Sciences

Antiderivatives and definite integrals, infinite series, applications to probability and dynamical systems.

#### UBC-MATH 104/184: Differential Calculus for Social Science and Commerce

Derivatives and rates of change, exponential and trigonometric functions, Newton's method, Taylor polynomials, maxima and minima, and graphing.

#### UBC-MATH 105: Integral Calculus with Applications to Commerce and Social Sciences

Antiderivatives, the definite integral, techniques of integration, infinite series, partial derivatives, maxima and minima with constraints, discrete and continuous random variables.

#### UBC-MATH 110: Differential Calculus

Topics as for MATH 100, but including relevant topics from algebra, geometry, functions, trigonometry, logarithms, and exponentials.

#### UBC-MATH 120: Honours Differential Calculus

Limits, derivatives, Mean Value Theorem and applications, elementary functions, optimization, Taylor series, approximation.

#### UBC-MATH 121: Honours Integral Calculus

Definite integrals and the Fundamental Theorem of Calculus, techniques and applications of integration, infinite series.

#### UBC-MATH 180: Differential Calculus with Physical Applications

Topics as for Math 100; intended for students with no previous knowledge of Calculus.

#### UBC-MATH 190: Calculus Survey

Functions, derivatives, integrals, curve sketching growth functions, volume calculations.

#### UBC-MATH 200: Calculus III

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.

#### UBC-MATH 210: Introduction to Mathematical Computing

Introduction to numerical computation, computer algebra, mathematical graphics. Primarily for second year students taking a degree in mathematics. One hour laboratory each week.

#### UBC-MATH 215: Elementary Differential Equations I

First-order equations; linear equations; linear systems; Laplace transforms; numerical methods; trajectory analysis of plane nonlinear systems. Applications of these topics will be emphasized.

#### UBC-MATH 217: Multivariable and Vector Calculus

Partial differentiation, extreme values, multiple integration, vector fields, line and surface integrals, the divergence theorem, Green's and Stokes' theorems.

#### UBC-MATH 220: Mathematical Proof

Sets and functions; induction; cardinality; properties of the real numbers; sequences, series, and limits. Logic, structure, style, and clarity of proofs emphasized throughout.

#### UBC-MATH 221: Matrix Algebra

Systems of linear equations, operations on matrices, determinants, eigenvalues and eigenvectors, diagonalization of symmetric matrices.

#### UBC-MATH 223: Linear Algebra

Matrices, eigenvectors, diagonalization, orthogonality, linear systems, applications.

#### UBC-MATH 226: Advanced Calculus I

Functions of several variables: limits, continuity, differentiability; implicit functions; Taylor's theorem; extrema; Lagrange multipliers; multiple integration, Fubini's theorem; improper integrals.

#### UBC-MATH 227: Advanced Calculus II

Parametrization of curves and surfaces; line and surface integrals; theorems of Green, Gauss, Stokes; applications to physics and/or introduction to differential forms.

#### UBC-MATH 230/335: Finite Mathematics (Math for future element. school teachers)

Difference equations, number theory, counting.

#### UBC-MATH 253: Multivariable Calculus

Partial and directional derivatives; maxima and minima; Lagrange multipliers and second derivative test; multiple integrals and applications. Equivalency MECH 222.

#### UBC-MATH 255: Ordinary Differential Equations

Review of linear systems; nonlinear equations and applications; phase plane analysis; Laplace transforms; numerical methods.

#### UBC-MATH 257: Partial Differential Equations

Introduction to partial differential equations; Fourier series; the heat, wave and potential equations; boundary-value problems; numerical methods.

#### UBC-MATH 264: Vector Calculus for Electrical Engineering

Divergence, gradient, curl, theorems of Gauss and Stokes. Applications to Electrostatics and Magnetostatics.

#### UBC-MATH 265: Linear Differential Equations

Linear ordinary differential equations. Complex numbers, Laplace transforms, frequency reponse, resonance, step response, systems.

#### UBC-MATH 267: Mathematical Methods for Electrical and Computer Engineering

Fourier series and transforms, wave equation, d'Alembert's solution, modes. Discrete Fourier tranform. Recurrence relations, z-transform, generating functions, applications.

#### UBC-MATH 300: Introduction to Complex Variables

Functions of a complex variable, Cauchy-Riemann equations, elementary functions, Cauchy's theorem and contour integration, Laurent series, poles and residues.

#### UBC-MATH 301: Applied Analysis

Integrals involving multi-valued functions, conformal mapping and applications, analytic continuation, Laplace and Fourier transforms.

#### UBC-MATH 302: Introduction to Probability

Basic notions of probability, random variables, expectation and conditional expectation, limit theorems. Equivalency: UBC STAT 302

#### UBC-MATH 303: Introduction to Stochastic Processes

Discrete-time Markov chains, Poisson processes, continuous time Markov chains, renewal theory.

#### UBC-MATH 305: Applied Complex Analysis

Functions of a complex variable, Cauchy-Riemann equations, contour integration, Laurent series, residues, integrals of multi-valued functions, Fourier transforms.

#### UBC-MATH 308: Euclidean Geometry

Classical plane geometry, solid geometry, spherical trigonometry, polyhedra, linear and affine transformations. Linear algebra proofs are used.

#### UBC-MATH 309: Topics in Geometry

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.

#### UBC-MATH 310: Abstract Linear Algebra

Linear spaces, duality, linear mappings, matrices, determinant and trace, spectral theory, Euclidean structure.

#### UBC-MATH 312: Introduction to Number Theory

Euclidean algorithm, congruences, Fermat's theorem, applications. Some diophantine equations. Distribution of the prime numbers.

#### UBC-MATH 313: Topics in Number Theory

Topics chosen by the instructor. These might include: division algorithms, group theory, continued fractions, primality testing, factoring.

#### UBC-MATH 316: Elementary Differential Equations II

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.

#### UBC-MATH 318: Probability with Physical Applications

Random variables, discrete and continuous distributions. Random walk, Markov chains, Monte Carlo methods. Characteristic functions, limit laws.

#### UBC-MATH 320: Real Variables I

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.

#### UBC-MATH 321: Real Variables II

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.

#### UBC-MATH 322: Introduction to Group Theory

Groups, cosets, homomorphisms, group actions, p-groups, Sylow theorems, composition series, finitely generated Abelian groups.

#### UBC-MATH 323: Introduction to Rings and Modules

Rings, ideals, unique factorization, Euclidean rings, fields, polynomial rings, modules; structure theory of modules over a principal ideal domain.

#### UBC-MATH 335: Introduction to Mathematics

Intensive course with required tutorial. Combinatorics, probability, geometry and elementary number theory.

#### UBC-MATH 340: Introduction to Linear Programming

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.

#### UBC-MATH 341: Introduction to Discrete Mathematics

Introduction to ideas and methods of discrete mathematics and their application.

#### UBC-MATH 342: Algebra and Coding Theory

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.

#### UBC-MATH 344: Mathematical Game Theory

Introduction to mathematical game theory and its applications.

#### UBC-MATH 345: Applied Nonlinear Dynamics and Chaos

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.

#### UBC-MATH 358: Engineering Analysis

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.

#### UBC-MATH 360: Mathematical Modeling in Science

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.

#### UBC-MATH 361: Introduction to Mathematical Biology

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.

#### UBC-MATH 400: Applied Partial Differential Equations

Separation of variables, first order equations, Sturm-Liouville theory, integral transform methods.

#### UBC-MATH 401: Green's Functions and Variational Methods

Green's functions for partial differential equations. Calculus of variations. Eigenfunction expansions. Rayleigh-Ritz and finite element methods.

#### UBC-MATH 402: Calculus of Variations

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.

#### UBC-MATH 403: Stabilization and Optimal Control of Dynamical Systems

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.

#### UBC-MATH 405: Numerical Methods for Differential Equations

Interpolation, numerical integration, numerical solution of ordinary and partial differential equations. Practical computational methods emphasized and basic theory developed through simple models.

#### UBC-MATH 406: Variational and Approximate Methods in Applied Mathematics

Variational and Green's function methods for ordinary and partial differential equations, introduction to finite difference, finite element and boundary element methods.

#### UBC-MATH 412: Advanced Linear Algebra

Topics include decompositions of linear operators, multi linear algebra, bilinear forms, metric spaces.

#### UBC-MATH 414: Mathematical Demonstrations

Students will prepare material illustrating ideas and applications of mathematics and present it to audiences outside the University.

#### UBC-MATH 418: Probability

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.

#### UBC-MATH 419: Stochastic Processes

Random walks, Markov chains, branching processes, Poisson processes, continuous time Markov chains, martingales, Brownian motion.

#### UBC-MATH 420: Real Analysis I

Sigma-algebras, Lebesgue measure, Borel measures, measurable functions, integration, convergence theorems, Lp spaces, Holder and Minkowski inequalities, Lebesgue and/or Radon-Nikodym differentiation.

#### UBC-MATH 421: Real Analysis II

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.

#### UBC-MATH 422: Fields and Galois Theory

Field extensions, the Galois correspondence, finite fields, insolvability in radicals, ruler and compass constructions, additional topics chosen by instructor.

#### UBC-MATH 423: Topics in Algebra

Commutative algebra, algebraic geometry, algebraic number theory, Lie theory, homological algebra and category theory, or some other advanced topic in algebra.

#### UBC-MATH 424: Classical Differential Geometry

The differential geometry of curves and surfaces in three-dimensional Euclidean space. Mean curvature and Gaussian curvature. Geodesics. Gauss's Theorema Egregium.

#### UBC-MATH 425: Introduction to Modern Differential Geometry

Riemannian manifolds, tensors and differential forms, curvature and geodesics.

#### UBC-MATH 426: Introduction to Topology

General topology, combinatorial topology, fundamental group and covering spaces, topics chosen by the instructor.

#### UBC-MATH 427: Topics in Topology

Homology theory, homotopy theory, manifolds, and other topics chosen by the instructor.

#### UBC-MATH 428: Mathematical Classical Mechanics I

Newton's equation, conservation laws, the Euler-Lagrange equation; Hamilton's principle of least action, Hamilton's equations, Lagrangian mechanics on manifolds.

#### UBC-MATH 430: Special Topics in Analysis

The student should consult the Mathematics Department for the particular topics offered in a given year.

#### UBC-MATH 432: Special Topics in Algebra

The student should consult the Mathematics Department for the particular topics offered in a given year.

#### UBC-MATH 437: Number Theory

Divisibility, congruencies, Diophantine equations, arithmetic functions, quadratic reciprocity, advanced topics.

#### UBC-MATH 440: Complex Analysis

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.

#### UBC-MATH 441: Mathematical Modeling: Discrete Optimization Problems

Formulation of real-world optimization problems using techniques such as linear programming, network flows, integer programming, dynamic programming. Solution by appropriate software.

#### UBC-MATH 442: Optimization in Graphs and Networks

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.

#### UBC-MATH 443: Graph Theory

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.

#### UBC-MATH 444: Mathematical Research and Writing

Current research topics in pure and applied mathematics are explored at the undergraduate level. Technical communication and research skills are developed.

#### UBC-MATH 446: Topics in the History of Mathematics I

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.

#### UBC-MATH 450: Asymptotic and Perturbation Methods

Asymptotic expansions. Asymptotic evaluation of integrals; WKBJ methods. Regular and singular expansions. Boundary layer theory; matched asymptotic expansions. Multiple scale techniques.

#### UBC-MATH 462: Projects in Mathematical Biology

Development and analysis of mathematical models for complex systems in ecology, evolution, cell biology, neurophysiology, and other biological and medical disciplines.