This module aims to equip students with the calculus and linear algebra based mathematical methods needed in the analysis of many engineering systems
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 Examination | 80 |