# Module Overview

### Mathematical Methods

This module aims to equip students with the calculus and linear algebra based mathematical methods needed in the analysis of many engineering systems

MATH H4000

##### ECTS Credits

5

*Curricular information is subject to change

Limits and Series.

Properties of limits. Differentiability. Implicit differentiation. Derivative of an inverse function. Taylor’s theorem. Maclaurin series. Application to approximation schemes for solving non-linear equations and calculating integrals. Convergence of power series and the Ratio Test. Integration by parts. Reduction formulas. The Gamma function.

Calculus of Several Variables

Chain rule for partial derivatives. Cylindrical and spherical polar coordinates. Directional derivative. The nabla operator. Calculation of gradient of a scalar field and divergence and curl of a vector field. Double and Triple integrals. Change of variables.

Laplace Transforms

Review of Laplace transform solution of second-order linear differential equations, and systems of first-order differential equations, with constant coefficients. Applications to engineering systems. Solution of classical partial differential equations by separation of variables. Laplace transform solution of partial differential equations occurring in engineering.

Vectors

Vectors in two and three dimensions. Geometric interpretation. Norm of a vector. Dot Product. Lines and planes. Extension to higher dimensions.

Vector spaces.

Definition of a Vector Space. Basis. Dimension. Spanning set. Linear independence.

Inner Products

Inner products. Norms. Orthogonality. Inner product spaces.

Linear Transformations

Definition of a Linear Transformation. Rotations in 2D and 3D. Kernel and Range. Matrices as Linear Transformations. The four fundamental subspaces.

Matrices

Matrices. Revision of matrix algebra. Classes of matrices. Inverse of a matrix. Determinants.

Systems of Equations

Systems of Equations. Partial pivoting. Gaussian Elimination in the context of LU Decomposition, Thomas Algorithm. Uniqueness of solutions.

Numerical Linear Algebra

Condition of a matrix. Iterative methods for sparse systems: Jacobi, Gauss-Seidel, SOR.

Module Content & Assessment
Assessment Breakdown %
Other Assessment(s)20
Formal Examination80