Faculty of Arts and Science

Polynomials and rational functions, trigonometry, exponential and logarithmic functions, inequalities, rudiments of probability and counting.

Linear systems and matrices. Matrix algebra. Determinants. Vector geometry. Complex numbers. Markov chains and other applications.

Prerequisite(s): |
One of Pure Mathematics 30, Mathematics 30, Mathematics 0500, or [Applied Mathematics 30 and at least 75 percent standing in Athabasca University’s Mathematics 101] |

Differentiation of elementary functions, the chain and product rules, extrema problems, integration. Applications from management, humanities and the social sciences.

Prerequisite(s): |
One of Pure Mathematics 30, Mathematics 30, Mathematics 0500, or [Applied Mathematics 30 and at least 75 percent standing in Athabasca University’s Mathematics 101] |

Substantially Similar: |
Mathematics 1560 |

Note: |
Mathematics 1510 may not be counted toward the requirements for a major in Mathematics and is not suitable for students requiring more than one semester of Calculus. |

Functions. Limits. Continuity. Differentiation and integration of polynomial, rational, root, trigonometric, exponential, and logarithmic functions. Applications of derivatives. Curve sketching and optimization. Anti-derivatives. Change of variable. Definite integrals. Fundamental Theorem of Calculus.

Prerequisite(s): |
One of Pure Mathematics 30, Mathematics 30, Mathematics 0500, or [Applied Mathematics 30 and at least 75 percent standing in Athabasca University’s Mathematics 101] |

Recommended Background: |
Mathematics 31 and a blended grade of at least 75 percent in Pure Mathematics 30 or Mathematics 30 |

Substantially Similar: |
Mathematics 1510 |

Logic, proofs. Set theory. Relations and functions. Finite and countable sets. Induction. Examples of axiomatic mathematical theories.

Principles of Logic. Number Systems and Bases. Sets of real numbers: Integers, Rationals, Irrationals. Modular Arithmetic and applications. Divisibility, primes and elementary number theory.

Note: |
Students should not take Mathematics 2090 if they have received credit for Mathematics 2000 prior to enrolling in Mathematics 2090. |

Applications of integration; logarithmic, exponential and hyperbolic functions; inverse functions; inverse trigonometric and hyperbolic functions; indeterminate forms; improper integrals; techniques of integration.

Prerequisite(s): |
Mathematics 1560 |

Sequences and series, convergence tests, Taylor’s series, vector-valued functions of a real variable, polar coordinates, applications to analytic geometry.

Prerequisite(s): |
Mathematics 1410;Mathematics 2560 |

Calculus of functions of several variables: partial differentiation, chain rule, applications, multiple integration, change of variables, theorems from vector analysis, including Stokes’ Theorem.

Prerequisite(s): |
Mathematics 2570 |

Graphs, trees and digraphs. Network flows. Scheduling. Enumeration, including the principle of Inclusion-Exclusion and generating functions.

Prerequisite(s): |
Mathematics 1410 |

Substantially Similar: |
Computer Science 1820 |

First Order Logic. Validity, provability, completeness, consistency, independence, categoricity, decidability, Gödel’s Theorem.

Prerequisite(s): |
Mathematics 2000 |

Introduction to classical geometry from the axiomatic point of view. Lines and affine planes. Separation, order, similarity, congruence. Isometries and their classification. Groups of symmetries. Projective, hyperbolic and inversive geometries.

Prerequisite(s): |
Mathematics 2000 |

Groups, abelian groups, subgroups, quotient groups. Homomorphism. Isomorphism theorems. Lagrange’s theorem. Permutation groups. Sylow theorems. Commutative rings, subrings, ideals. Quotient rings and ideals. Polynomial rings.

Prerequisite(s): |
Mathematics 2000 |

Vector spaces over the real and complex numbers. Basis and dimension. Linear transformations. Change of basis. Gram-Schmidt orthogonalization. Eigenvectors and diagonalization. Canonical forms. Cayley-Hamilton Theorem.

Prerequisite(s): |
Mathematics 1410;Mathematics 2000 |

Division algorithm. Fundamental Theorem of Arithmetic. Euclidean Algorithm. Linear Diophantine equations. Congruences. Chinese Remainder Theorem. Quadratic reciprocity. Additional topics such as Pythagorean triples, Gaussian integers, sums of squares, continued fractions, arithmetic functions, or cryptography.

Prerequisite(s): |
Mathematics 2000 |

Rigorous treatment of the notions of calculus of a single variable, emphasizing epsilon-delta proofs. Completeness of the real numbers. Upper and lower limits. Continuity. Differentiability. Riemann integrability.

Prerequisite(s): |
Mathematics 2000;Mathematics 2570 |

Complex number system and complex plane. Analytic functions. Complex integration. Power series. Calculus of residues.

Prerequisite(s): |
Mathematics 2580;One of Mathematics 2000 or Physics 2150 |

First order ordinary differential equations. Second and higher order ordinary differential equations. Linear systems of ordinary differential equations. Qualitative theory of ordinary differential equations. Applications. Series solutions. Singular point expansions. Elementary linear difference equations.

Prerequisite(s): |
Mathematics 1410;Mathematics 2560 |

Corequisite(s): |
Mathematics 2570 |

Burnside’s theorem, Polya’s theorem. Finite fields and combinatorial design. Coding theory. Ramsey Theory.

Prerequisite(s): |
Mathematics 2865 |

Topological spaces. Topology of metric spaces. Continuity. Open covers and compactness. Separation. Connectedness.

Prerequisite(s): |
Mathematics 3500 |

Polynomial rings. Fields and field extensions, construction problems. Finite fields. Galois Theory. Fundamental Theorem of Algebra.

Prerequisite(s): |
Mathematics 3400 |

This follow-up course to Mathematics 3500 (Analysis I) presents concepts that are crucial for an understanding of both pure and applied mathematics at an advanced level. Sequences and series of functions. Uniform continuity. Uniform convergence. The Stone-Weierstrass Theorem. The Lebesgue (or Riemann-Stieltjes) integral. Fourier series. Other topics.

Prerequisite(s): |
Mathematics 3500 |

Adjoints. Oscillation theory. Matrix methods. Matrix exponential functions. Sturm-Liouville theory. Orthonormal systems and Fourier series. Eigenfunction expansions. Laplace, Fourier and Mellin transforms. Convolutions. Convergence theory. Plancherel and Parseval formulae. Distributions. Solving PDEs using integral transforms. Fundamental solutions. Separation of variables. Heat, wave and Poisson equations. Harmonic functions.

Prerequisite(s): |
Mathematics 2570;Mathematics 3600 |

Recommended Background: |
Mathematics 3500 |