M＆S-UWI

 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.

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.

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.

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.