| 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 Cramers rule. Eigenvalues and eigenvectors. Diagonalization of matrices. Cayley-Hamilton theorem. Linear Independence, basis and dimension, with connection to row operations and determinant. |