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This subject introduces the field of mathematical analysis both with a careful theoretical framework as well as selected applications. Many of the important results are proved rigorously and students are introduced to methods of proof such as mathematical induction and proof by contradiction.
The important distinction between the real numbers and the rational numbers is emphasised and used to motivate rigorous notions of convergence and divergence of sequences, including the Cauchy criterion. Various constructions of the real numbers, for example using Dedekind cuts or by completion, are discussed and shown to be equivalent. These ideas are extended to cover the theory of infinite series, including common tests for convergence and divergence. Compactness of the unit interval is established and various consequences of compactness, such as the Extreme Value Theorem, are discussed. A similar treatment of continuity and differentiability of functions of a single variable leads to applications such as the Mean Value Theorem and Taylor’s theorem. We define and compare both the Lebesgue and Riemann integral, establish basic properties of both, and dis- cuss the proof of the Fundamental Theorem of Calculus. The convergence properties of sequences and series are explored, with applications to power series representations of elementary functions and their generation by Taylor series. Fourier series are introduced as a way to represent periodic functions. Further topics may include: uniform continuity, equicontinuity, the Arzela-Ascoli theorem, and the Stone-Weierstrass theorem.
Intended learning outcomes
On completion of this subject, students should be able to:
- Apply rigour in mathematics, be able to use proof by induction, proof by contradiction, and to use epsilon-delta proofs both as a theoretical tool and a tool of approximation;
- Describe the theory and applications of both the Riemann and Lebesgue integrals;
- Identify real numbers and explain their importance of completeness;
- Define convergence and the Cauchy property, and be able to deploy either when analysing the convergence or divergence of sequences;
- Apply knowledge of the theory and practice of power series expansions and Taylor polynomial approximations;
- Describe the role of Fourier series in representing periodic functions.
problem-solving skills: the ability to engage with unfamiliar problems and identify relevant solution strategies;
analytical skills: the ability to construct and express logical arguments and to work in abstract or general terms to increase the clarity and efficiency of analysis;
collaborative skills: the ability to work in a team;
time management skills: the ability to meet regular deadlines while balancing competing commitments.
Last updated: 15 September 2023