M＆S-UWI

The course covers the three sections of

1. Complex numbers,
2. Analysis, and
3. Matrices.

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.

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.

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.

This course is an introduction to complex analysis, or the theory of the analytic functions of a complex variable. Such functions, beautiful on their own, are immediately useful in Physics, Engineering, and Signal Processing. However, the concepts introduced in complex analysis are nowadays also ubiquitous in number theory and combinatorics.

Students will calculate line integrals, and the techniques they will learn in this course will help them get past many of the seeming “dead ends” they ran up against in calculus. Indeed, most of the definite integrals they will learn to evaluate cannot be solved except through techniques from complex analysis.

Mathematics is a powerful tool used for solving practical problems and is a highly creative field of study, combining logic and precision with intuition and imagination. The ability to employ mathematical reasoning is a fundamental skill for any well-educated individual in the pure and applied sciences. 

This preliminary course is designed to provide a solid foundation for students interested in further studies in mathematics and also to give students in the Sciences the mathematical tools necessary for their work. It is a prerequisite for students who intend to take advanced level courses in mathematics or statistics.

This course serves to sharpen the analytical and critical reasoning skills of the student and to improve his/her ability to express mathematical ideas with coherence and clarity.

Linear algebra is concerned with the study of vectors, vector spaces, and linear transformations between vector spaces. Linear algebra has among its many applications a basic connection with coordinate geometry. For instance, lines, planes, and conic sections may be analysed this way.

In this course, the geometric interpretation of vectors and the operations on and between them (e.g., the scalar and vector products) is emphasized. Geometrical properties of lines and planes, possible intersection points and distances are explored. We also look at conic sections, and transformations of geometrical objects giving a geometrical interpretation of the determinant of a matrix.

Matrices also appear in the study of systems of linear equations which are solved by Gaussian elimination or Cramer’s rule. Matrix operations as well as the recursive definition of determinants are explained.

As a by-product, we also study the geometry of complex number.

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 covers 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. A sound knowledge of the required prerequisite material (CAPE Pure Mathematics Units 1 & 2 or Preliminary Mathematics 1 & 2) is assumed. While there is a considerable overlap with the ideas encountered in the prerequisite courses, 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 and applications thereof, e.g., finding maxima and minima of functions. Integrals are introduced via Riemann sums and methods for their evaluation using antiderivatives are considered.

The availability of computer algebra systems has been changing the mathematical landscape considerably in the past few years: Real-world problems that were deemed computationally too long-winded or complicated can now be tackled. Appropriate visualizations help interpret the results obtained, mathematical software can be used to “experiment” with mathematical structures and find connections between them.

In this course, students are provided with hands-on access to mathematics. They are introduced to the modern mathematical software package Sage and the statistical software package R via the cloud-service “CoCalc”, as well as the intuitive geometry program GeoGebra.

Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving multiple variables, rather than just one. Techniques of multivariable calculus are used to study many objects of interest in the material world like curves, surfaces and vector fields.

Ordinary differential equations (ODEs) arise in many contexts of mathematics and science (social as well as natural). Mathematical descriptions of change use differentials and derivatives. Various differentials, derivatives, and functions become related to each other via equations, and thus a differential equation is a result that describes dynamically changing phenomena, evolution, and variation. Often, quantities are defined as the rate of change of other quantities (for example, derivatives of displacement with respect to time), or gradients of quantities, which is how they enter differential equations.

This course continues, complements and deepens the study of algebraic ideas and concepts as begun in MATH2310 Abstract Algebra 1 (and thus these two courses are offered in consecutive semesters). Structures like groups, rings, and fields as well as mappings like homomorphisms and isomorphisms will be revisited in this course.

In this course, students will explore groups in greater depth; in particular, group actions are used to prove several key results about finite groups, and the famous Sylow Theorems are discussed. Students will be exposed to certain aspects used in the classification of finite simple groups, like the Decomposition Theorem and the Jordan-Hölder Theorem.

The course then progresses to topics in algebraic number theory where students learn to construct algebraic extensions containing the root of a polynomial. Several interesting and motivating examples like the straigthedge and compass constructions will be discussed.

This course exposes students to rigorous mathematical definitions of derivatives and (Riemann) integrals, together with topics like uniform convergence of a sequence of functions and series of functions. It continues and completes the study of the rigorous foundations of calculus begun in MATH2321 Real Analysis 1. A major emphasis is placed on the proper use of definitions for the rigorous proof of theorems.

This course is compulsory for all students pursuing a major in mathematics. Students interested in advanced analytical courses like complex analysis, Fourier analysis, numerical analysis will need a good grasp of the concepts discussed in this course.

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