Engineering
MA2102: Mathematics and Applied Mathematics II (5 ECTS)
This module considers topics in both Mathematics and Applied Mathematics for engineering students. The material covered includes linear algebra, sequences and series, complex analysis, dimensional analysis, and partial differential equations.
Taught in Semester(s) 2. Examined in Semester(s) 2.
Workload: 123 hours (36 Lecture hours, 12 Tutorial hours, 75 Self study hours).
Module Learning Outcomes.
On successful completion of this module the learner should be able to:
- identify vector spaces and subspaces and be able to carry out vector / matrix operations: to establish linear independence; find a basis for / dimension of a space and to apply linear transformations;
- calculate the eigenvalues & eigenvectors of a matrix, apply the diagonalisation technique, and implement the Gram-Schmidt process;
- find the limit of some sequences, and establish the convergence or divergence of some series, including some power series;
- test whether a given function of a complex variable can be analytic using the Cauchy-Riemann equations;
- simplify expressions governing engineering systems with the aid of dimensional analysis;
- design scale models with the aid of dimensional analysis;
- mathematically model some simple systems using partial differential equations;
- solve some problems involving partial differential equations using the method of separation of variables.
Indicative Content
This module covers material in both Mathematics and Applied Mathematics for engineering students, and the following topics are considered.
Linear algebra. Vector spaces and subspaces, vector and matrix operations, linear independence of vectors, a basis for a space, the dimension of a space, linear transformations. Eigenvalues & eigenvectors of a matrix, and the diagonalisation technique. Inner products, orthonormal bases and the Gram-Schmidt process. The least squares method.
Sequences and series. Convergence of sequences and series, convergence tests, power series.
Complex variable theory. Functions of a complex variable, limits, continuity and complex differentiation, analytic functions, the Cauchy-Riemann equations.
Dimensional analysis. Non-dimensional variables, the Buckingham Pi theorem, constructing a set of independent non-dimensional variables for a system, physical similarity and designing scale models.
Partial differential equations (PDEs). Three important examples: the heat equation, the wave equation, and Laplace’s equation, and how these equations arise as models of engineering systems. Boundary and initial conditions, well-posed problems. The solution of PDEs using the method of separation of variables. Solution interpretation in engineering contexts.
Module Resources
- Modern Engineering Mathematics, G. James, Prentice Hall
- Advanced Modern Engineering Mathematics, G. James, Prentice Hall
- Advanced Engineering Mathematics, E. Kreyszig, John Wiley & Sons
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