Module Descriptors

Mathematical Studies

MA301: Advanced Calculus (5 ECTS)

This calculus course builds on earlier basic calculus knowledge. Topics covered include: convergence of sequences & series, Taylor's & the Maclaurin series, multiple integrals using Cartesian, polar and elliptical coordinates, Fourier series, computation of line integrals directly and by using Green's theorem.

Taught in Semester(s) 1. Examined in Semester(s) 1.

Workload: 106 hours (24 Lecture hours, 12 Tutorial hours, 70 Self study hours).

Module Learning Outcomes. On successful completion of this module the learner should be able to:

  1. Define and describe a sequence and establish if a sequence converges.
  2. Define a series and establish if a series converges/diverges, converges absolutely/ conditionally. Define a geometric, telescopic and the harmonic series. Use the integral test and in particular use it to find which values of p for which the p series converges. Apply the comparison test, ratio test and root test.
  3. Define a general Taylor and Maclaurin series. Compute the coefficients of the power series and establish the centre, radius and interval of convergence. Evaluate approximately a function at various points using power series.
  4. Define a Fourier series, even and odd functions and compute Fourier coefficients.
  5. Compute volumes under surfaces using double integrals over rectangler and non-rectangler domains.
  6. Use polar and elliptical coordinates to compute volumes over full/segments of circular/elliptical domains.
  7. Compute line integrals over curves in the Euclidian 2 space directly and by using Green's theorem.

Indicative Content

This course builds on earlier basic calculus. The material covered includes:

  1. Sequences & series: Sequence definition/description, terms used to describe sequences e.g upper bound, limits, convergence of a sequence, definition of a series, convergence of a series, absolute/conditional convergence, geometric series, telescopic series, analysis of the harmonic series, the integral test, p series, comparison test, ratio test, root test.
  2. Power series: Definition of a general power series, centre of convergence, radius of convergence, interval of convergence including end points, coefficients of the power series, Taylor series, Maclaurin series, approximate evaluation of functions at various points using power series, definition of a Fourier series, odd and even functions, period of a function, computation of the Fourier series coefficients.
  3. Double integration: comparison with single variable integration, double integral as a volume under a surface, evaluation of double integrals using known volumes under surfaces, evaluation of double integrals using integration techniques over rectangular and non-rectangular domains of integration.
  4. Polar and elliptical coordinates: definition of polar and elliptical coordinates, Jacobian determinant, evaluation of volumes under surfaces over full/partial elliptical and circular domains, double integrals used to compute areas of domains.
  5. Line integrals: paramaterisation of curves in Euclidean 2 space, chain rule, integration techniques, Green's theorem, evaluation of a line integral using Green's theorem.

Module Resources

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