AUMAT - Augustana Faculty - Mathematics

Review of the mathematical background essential to success in Elementary Calculus I, as well as an introduction to some of the central concepts of calculus. Review topics include polynomials, rational expressions, exponents and logarithms, the real number line, the Cartesian plane, trigonometry, and functions and their graphs. Prerequisites: Mathematics 30-1. Note: Normally offered as a nine-week course in the latter portion of the first term. Students with unsatisfactory performance through the first four weeks of AUMAT 110 may be permitted to withdraw from that course and register in the next offering of AUMAT 101. Students obtaining credit in in AUMAT 101 are strongly encouraged to attempt the next offering of AUMAT 110. Not open to students with credit in AUMAT 110 or 116, and normally not open to a student with credit in Mathematics 31. The course does not count toward the major in Mathematics and Physics or the minor in Mathematics.

Elementary number theory, numeration systems, number systems, sets, logic, and elementary probability theory. Prerequisite: Mathematics 30-1 or 30-2, or consent of the instructor. Notes: The course does not count toward the major in Mathematics and Physics or the minor in Mathematics, nor may it be used for credit towards a B.Sc. degree. Credit may not be obtained for AUMAT 107 if credit has already been obtained for AUMAT 250.

Limits; differentiation and integration of algebraic, trigonometric, exponential, and logarithmic functions; applications. Prerequisite: Mathematics 30-1. Notes: Credit may be obtained for only one of AUMAT 110 and 116. Students with credit in Mathematics 31 who score 80% or more on the Calculus Placement Test should take AUMAT 116 instead of AUMAT 110. Students with unsatisfactory performance through the first four weeks of the course are advised to withdraw and register in the next offering of AUMAT 101.

Fundamental Theorem, inverse trigonometric functions and their derivatives, indeterminate forms, improper integrals, techniques of integration, applications. Prerequisite: AUMAT 110 or 116.

Foundational topic of calculus (including differentiation and integration of algebraic, trigonometric, exponential and logarithmic functions; Fundamental Theorem) with a focus on modelling and elementary differential equations. Prerequisite: Mathematics 30-1. Note: Credit may be obtained for only one of AUMAT 116 or 110.

Vector and matrix algebra, determinants, linear systems of equations, vector spaces, eigenvalues and eigenvectors, applications. Prerequisite: Mathematics 30-1.

Infinite series, plane curves, polar coordinates, vectors and three-dimensional analytic geometry, cylindrical and spherical coordinates, elements of linear differential equations. Prerequisite: AUMAT 112.

Functions of several variables, partial derivatives, integration in two and three dimensions, vector functions, space curves, arc length, line integrals, Green's theorem, surface integrals, Stokes' theorem, the divergence theorem. Prerequisite: AUMAT 211.

Further foundational topics in calculus, including: limits of sequences and functions, infinite series (including Taylor and Fourier series) and multi-variable differential calculus. Prerequisites AUMAT 110 or 116, and AUMAT 120.

Groups as a measure of symmetry. Groups of rigid motions. Frieze groups, and finite groups in two and three dimensions. Groups of matrices. Group actions with application to counting problems. Permutation groups. Subgroups, cosets, and Lagrange's Theorem. Quotient groups and homomorphisms. Prerequisites: AUMAT 120 and one of 110 or 116.

Mathematical analysis of problems arising in economics and finance, including an introduction to economic modelling; simple, compound, and continuous rates of interest; static and comparative-static analysis; optimization; annuities, mortgages, bonds, and other securities; and dynamics. Prerequisites: AUECO 101 and one of AUMAT 110 or 116. Note: Credit may be obtained for only one of AUMAT 235, AUECO 206, AUMGT 206.

Computational methods and software packages and libraries in the mathematical sciences with applications to differentiation and integrations, data fitting, nonlinear systems and differential equations. Prerequisites: AUCSC 111 or AUCSC 113 (2023), and AUMAT 110 or 116; or consent of the instructor. Corequisite: AUMAT 120. Note: Credit may be obtained for only one of AUMAT 240, AUSCI 250 (2023), AUMAT 340 (2020), AUCSC 340 (2020), or AUPHY 340 (2020).

Sets, functions, elementary propositional and predicate logic, Boolean algebra, elementary graph theory, proof techniques (including induction and contradiction), and combinatorics. Prerequisites: AUMAT 110 or 116, and 120.

Axiomatic systems and finite geometries; neutral geometry and the various parallel postulates, leading to Euclidean and hyperbolic geometry; constructions; isometries of the plane and groups of transformations, and inversions in circles; models for Euclidean and hyperbolic geometry; applications. Prerequisite: AUMAT 120 or consent of the instructor.

Complex numbers, functions of a complex variable, analytic functions, Cauchy and related theorems, Taylor and Laurent expansions, the residue calculus and applications, harmonic functions, conformal mapping, applications. Prerequisite: AUMAT 212.

Senior topics in calculus and mathematical methods in physics. Topics include: coordinate systems; applications of single integrals; multiple integrals and applications; vector analysis; ordinary differential equations; partial differential equations; complex numbers. Prerequisites: AUMAT 216.

Computational problems in linear algebra, including linear systems and least squares and eigenvalues problems, with matrix factorizations as a main tool for tackling these problems. Prerequisites: AUMAT 240 or AUSCI 250 (2023).

Introduction to cryptography in theory and practice, including its applications and mathematical foundations. Topics include classical cryptosystems, private-key cryptosystems (including DES and AES), hashing and public-key cryptosystems (including RSA), digital signatures, selected topics in cryptography. Prerequisite: AUMAT 250 and one of AUMAT 240 or AUSCI 250 (2023).

First- and higher-order equations; methods of solution, including complex variable techniques; series solutions; elementary transform techniques; oscillation theory; applications to biology and physics. Prerequisite: AUMAT 120, 211.

Fundamental concepts of discrete and continuous dynamical systems, both linear and nonlinear; nonlinear differential equations; deterministic, nondeterministic, and chaotic dynamics; strange attractors and fractals. Applications in ecology, biology and physics. Prerequisites: AUMAT 216.

Introduction to optimization (definition, notation and taxonomy); unconstrained optimization using gradient descent and stochastic gradient descent; linear programming: The Simplex Method; constrained optimization and Lagrange multipliers; convex optimization and quadratic programming. Prerequisites: AUMAT 216 and one of AUMAT 240 or AUSCI 250 (2023).

Introduction to elementary probability theory and stochastic processes with a special emphasis on their applications in science. Topics include basics of probability, random variables, functions of random variables, random vectors, random processes and their classification, well-known random processes including the Bernoulli process, random walk process, Gaussian process, Poisson process, and Markov process. Prerequisite: AUMAT 250.

Intensive study of a specific mathematical problem or other area of mathematics as defined by the student and a supervising instructor. Notes: Admission to AUMAT 395 normally requires a minimum GPA of 3.0 on the major in Mathematics and Physics. An Application for Individual Study must be completed and approved before registration in the course.

Integrated history of mathematics and physics, emphasizing the scientific revolution and the subsequent development of mathematics and physics as distinct disciplines. Prerequisite: AUMAT 211 and one of AUMAT 220, 229, 250. Note: Credit may be obtained for only one of AUMAT 480 or AUPHY 480.

Intensive study of a specific problem or area of mathematics as defined by the student and a supervising instructor. Prerequisite: Fourth-year standing. Notes: Admission to AUMAT 495 normally requires a minimum GPA of 3.0 on the major in Mathematics and Physics. An Application for Individual Study must be completed and approved before registration in the course.