Courses :: Module, MATHEMATICS II

	Overview
	Indicative Syllabus • Introduction, motivation and scope • The Laplace Transform • Basic Laplace Transform • Inverse Laplace Transform • Solving first-order ODEs using the Laplace Transform. • Vector algebra • Vectors in coordinate independent form and in 2-Space, 3-Space and n-Space. • Addition, multiplication of vectors by a scalar. • Basic properties of vectors, subtraction and scalar (dot) and vector (cross) product of vectors. • Rules of Vector algebra. • Unit vectors i,j,k. • Inner (Dot) product in n-space and Vector (Cross) product in 3space. • Orthogonality of vectors. • Applications of inner and vector products. • Vector equation of a line in 3-space, Equation of plane in 3-space. • Triple scalar product, Triple vector product. • Time derivative of a vector function. • Vector equation of a curve in parametric form in 2 and 3 space. • Arc length, unit tangent, unit principal normal vectors. Curvature. • Matrix algebra • Overview of Matrix algebra and its usefulness in solving systems of linear equations. • Matrix addition, subtraction and multiplication. • Matrix properties – transpose, symmetric, rank • Solving linear systems of equations using Gaussian and Gauss-Jordan elimination. • Special matrices such as the identity matrix, the zero matrix, and orthogonal matrices. • Solving linear systems using the inverse of a matrix. Properties of the inverse matrix. • Calculating the inverse matrix using row operations. • Definition of the determinant and discussion of its properties. • Calculating the inverse matrix using the method of cofactors. • Solving linear systems using the inverse matrix and Cramer’s rule. • Eigenvalues and eigenvectors. • Diagonalization of matrices. Cayley-Hamilton theorem. • Linear Independence, basis and dimension, with connection to row operations and determinant.

Overview

Indicative Syllabus • Introduction, motivation and scope • The Laplace Transform • Basic Laplace Transform • Inverse Laplace Transform • Solving first-order ODEs using the Laplace Transform. • Vector algebra • Vectors in coordinate independent form and in 2-Space, 3-Space and n-Space. • Addition, multiplication of vectors by a scalar. • Basic properties of vectors, subtraction and scalar (dot) and vector (cross) product of vectors. • Rules of Vector algebra. • Unit vectors i,j,k. • Inner (Dot) product in n-space and Vector (Cross) product in 3space. • Orthogonality of vectors. • Applications of inner and vector products. • Vector equation of a line in 3-space, Equation of plane in 3-space. • Triple scalar product, Triple vector product. • Time derivative of a vector function. • Vector equation of a curve in parametric form in 2 and 3 space. • Arc length, unit tangent, unit principal normal vectors. Curvature. • Matrix algebra • Overview of Matrix algebra and its usefulness in solving systems of linear equations. • Matrix addition, subtraction and multiplication. • Matrix properties – transpose, symmetric, rank • Solving linear systems of equations using Gaussian and Gauss-Jordan elimination. • Special matrices such as the identity matrix, the zero matrix, and orthogonal matrices. • Solving linear systems using the inverse of a matrix. Properties of the inverse matrix. • Calculating the inverse matrix using row operations. • Definition of the determinant and discussion of its properties. • Calculating the inverse matrix using the method of cofactors. • Solving linear systems using the inverse matrix and Cramer’s rule. • Eigenvalues and eigenvectors. • Diagonalization of matrices. Cayley-Hamilton theorem. • Linear Independence, basis and dimension, with connection to row operations and determinant.

	Learning Outcomes
	On successful completion of the module, students should be able to: carry out basic vector operations and algebraically reduce multiple vector products to their simplest form; obtain the unit tangent vector, unit principal normal vector and curvature of curves in 2 and 3-space and Perform basic arithmetical and special operations with matrices; write a system of linear equations in matrix form and to use row elimination methods to solve the system; determine the nature of the solutions to a linear system by examination of the properties of the coefficient matrix; calculate the eigenvalues and eigenvectors of a matrix and determine matrix properties such as linear independence, rank, nullity, basis and dimension; determine the Laplace transforms of elementary functions; solve simple first-order ODEs using the Laplace transform; apply and select the appropriate mathematical techniques to solve a variety of associated engineering problems.

Learning Outcomes

	Autumn Supplementals/Resits
	Are supplemental/resit registrations permitted? Yes The continuous assessment mark is carried forward to the Autumn examinations as there is no facility available for repeating the continuous assessment components of the course. University scheduled written examination (Autumn): 120 minutes, On-campus

Autumn Supplementals/Resits

	Timetable
	This timetable is not currently available, please contact the International Engineering College Department.

Timetable

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MATHEMATICS II