- Lecturer: Duaine Lewis
Calculus and Analytical Geometry
- Complex numbers,
- Analysis,
and
- Matrices.
- Lecturer: Bernd Sing
Sets and Number Systems
To fully understand mathematics, the essential rules of logic for the formulation of mathematical proofs have to be studied. The various types of mathematical proofs are discussed, mainly using examples in elementary set theory and number systems.
In this course, students are introduced to the basic operations that are performed on sets. In so doing the concepts of relations, functions and binary operations are examined. Next we examine natural numbers and the derived principle of mathematical induction. We discuss permutations, combinations and sequences that are defined recursively with respect to natural numbers. The properties of the integers such as divisibility, greatest common divisor and Euclidean algorithm are examined. We study the field axioms in relation to rational numbers. We solve linear and non-linear inequalities involving the real numbers.
- Lecturer: Duaine Lewis
Calculus B
In a broad sense, calculus is the mathematical study of change consisting of differential calculus (concerning rates of change and slopes of curves) and integral calculus (concerning accumulation of quantities and the area under/between curves).
The course continues the study of the fundamental concepts of the differential and integral calculus of a single variable. The used approach is intuitive and informal but nevertheless prepares students to understand the rigorous proofs at a subsequent level. While there is a considerable overlap with the ideas encountered in Calculus A, a greater emphasis is put on the fundamental ideas involved in differentiation and integration. The main results are derived and illustrated although often heuristic proofs are used.
This course will treat the limits, continuity and differentiability of a function of a single variable from a more rigorous point of view. Double integrals are introduced and methods for their evaluation are considered.
- Lecturer: Bernd Sing
Real Analysis 1
This course exposes students to rigorous mathematical definitions of
limits of sequences of numbers and functions, classical results about
continuity and series of numbers and their proofs. A major emphasis is
placed on the proper use of definitions for the rigorous proof of
theorems. The following topics will be covered: The real number system,
topological properties of real numbers, sequences, continuity and
uniform continuity.
- Lecturer: Bernd Sing
Metric Spaces
This is a first course in point set topology. The subject of topology grew out of the study of geometric and analytic properties of the real line and Euclidean space. In particular, topology studies generalizations of the concepts of union, intersection, open intervals, closed intervals, limit points, and continuous functions. The material of topology is a combination of ideas from algebra, analysis and geometry.
In this course on metric spaces, topological concepts already met in MATH2321: Real Analysis 1 and MATH3550: Real Analysis 2 will be discussed in greater detail and generality. Besides the necessary abstract theory, applications are also pointed out, e.g., Picard’s Theorem that is so fundamental in the study of differential equations, or the how GIS (geographic information system) stores information about the relationship between spatial regions. Thus this course on metric spaces could be considered an applied topology course.
This course is compulsory for all students majoring in mathematics. It is also of interest to students that deal with objects and data in higher-dimensional spaces.