**MATHEMATICS 0500
Essential Mathematics**

Contact hours per week: 3-0-1

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

*Prerequisite: Applied Mathematics 30 or equivalent.*

**Note:** *This course may not be taken for credit by students with Pure Mathematics 30 or equivalent.*

*This course may not be included among the mathematics courses required for Computer Science or Mathematics majors in Arts and Science.*

Elementary Linear Algebra

Contact hours per week: 3-0-1

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

*Prerequisite: Mathematics 30 or Pure Mathematics 30, Mathematics 0500, or Applied Mathematics 30 and at least 75% standing in Athabasca University's Mathematics 101.*

Techniques of Calculus

Contact hours per week: 3-0-1

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

*Prerequisite: Mathematics 30 or Pure Mathematics 30, Mathematics 0500, or Applied Mathematics 30 and at least 75% standing in Athabasca University's Mathematics 101.*

**Note:** *Credit is not allowed for Mathematics 1510 and 1560.*

*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.*

Calculus I

Contact hours per week: 3-0-1

Functions, limits, continuity, differentiation of elementary functions, trigonometric functions and their derivatives, applications of derivatives, curve sketching and optimization, anti-derivatives, change of variable, definite integrals, the fundamental theorem of calculus.

*Prerequisite: Mathematics 30 or Pure Mathematics 30, Mathematics 0500, or Applied Mathematics 30 and at least 75% standing in Athabasca University's Mathematics 101.*

*Recommended background: Mathematics 31 and a blended grade of at least 75 percent in Mathematics 30 or Pure Mathematics 30.*

**Note:** *Credit is not allowed for Mathematics 1510 and 1560.*

Mathematical Concepts

Contact hours per week: 3-0-1

Logic, proofs. Set theory. Relations and functions. Finite and countable sets. Induction. Number theory. Elementary group theory.

*Prerequisite: Mathematics 1410.*

*Recommended background: Mathematics 1560.*

Number Systems

Contact hours per week: 3-0-1

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

*Prerequisite: Completion of eight university-level courses (24.0 credit hours).*

**Note:** *Credit is not allowed for Mathematics 2090 and Mathematics 2000. Mathematics 2090 may not be counted toward the requirements for a major in Mathematics or Computer Science. Mathematics 2090 is primarily intended for prospective elementary school teachers who would not ordinarily take university math courses.*

Calculus II

Contact hours per week: 3-0-1

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

*Prerequisite: Mathematics 1560.*

Calculus III

Contact hours per week: 3-0-0

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

*Prerequisites: Mathematics 1410 and 2560.*

Calculus IV

Contact hours per week: 3-0-0

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

*Prerequisite: Mathematics 2570.*

Combinatorial Mathematics

Contact hours per week: 3-0-0

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

*Prerequisite: Mathematics 1410.*

Introduction to Mathematical Logic

Contact hours per week: 3-0-0

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

*Prerequisite: Mathematics 2000.*

**Note:** Credit is not allowed for Mathematics 3100 and Logic 3003.

Geometry

Contact hours per week: 3-0-0

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: Mathematics 2000.*

Topology

Contact hours per week: 3-0-0

Introduction to general topology. Continuity, connectedness, compactness, separation.

*Prerequisites: Mathematics 2000 and 2560.*

*Recommended background: Mathematics 3500.*

Group and Ring Theory

Contact hours per week: 3-0-0

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: Mathematics 2000.*

Linear Algebra

Contact hours per week: 3-0-0

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: Mathematics 1410.*

*Corequisite: Mathematics 2000.*

Elementary Number Theory

Contact hours per week: 3-0-0

Integral domains, modular arithmetic, Gaussian integers, divisibility, primes, congruences, Chinese remainder theorem, quadratic reciprocity, sums of squares, diophantine equations.

*Prerequisite: Mathematics 2000.*

Analysis

Contact hours per week: 3-0-0

Detailed examination of the ideas and results of the first courses in integral and differential calculus. Countable and uncountable sets, properties of the space of real numbers, sequences, series and convergence. Continuity, differentiability, integrability.

*Prerequisites: Mathematics 2000 and 2560.*

Differential Equations I

Contact hours per week: 3-0-0

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. Emphasizes techniques and methods.

*Prerequisites: Mathematics 1410 and 2560.*

*Corequisite: Mathematics 2570.*

Topics in Mathematics

Contact hours per week: 3-0-0

Combinatorics

Contact hours per week: 3-0-0

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

*Prerequisite: Mathematics 2865.*

Optimization

Contact hours per week: 3-0-0

Networks. Linear programming, including duality and the simplex method. Integer linear programming methods. Assignment problem, knapsack problem, critical path scheduling. Introduction to non-linear programming and search methods.

*Prerequisites: Mathematics 2000 and 2865.*

Field Theory

Contact hours per week: 3-0-0

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

*Prerequisite: Mathematics 3400.*

Complex Analysis

Contact hours per week: 3-0-0

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

*Prerequisite: Mathematics 2570.*

*Recommended background: Mathematics 3500.*

Differential Equations II

Contact hours per week: 3-0-0

Theoretical foundations. Integral transforms. Discontinuous coefficients. Separation of variables. Fourier series. Linear partial differential equations of order 2. Boundary value problems and Sturm-Liouville Theory. Applications. Distribution Theory.

*Prerequisites: Mathematics 2570 and 3600.*

*Recommended background: Mathematics 3500.*

Topics in Mathematics

Contact hours per week: 3-0-0