This course has the objective of gaining a solid background in applied mathematics as related to various research areas of science and engineering.
Matrices and Vector Spaces: This part of the course focuses on reviewing the concepts of linear vector spaces, vector operations, linear operators, matrix definition and properties. It will also cover the definition of eigenvalues and eigenvectors, and solution methods to simultaneous linear equations.
Vector Calculus: In this unit, differentiation of scalar, vector and tensor fields will be covered. Concepts of gradient, divergence, curl and Laplacian will be introduced and applied to different coordinate systems, such as, Cartesian, polar, cylindrical and spherical coordinates
Line, Surface and Volume Integrals: The aim of this section is to develop methods for handling multi-dimensional physical situations. Line, surface and volume integrals definitions will be used to define conservative fields, scalar potentials. In addition, the divergence theorem and the Stoke’s theorem will be introduced.
Ordinary Differential Equations: In this section, solution of both homogeneous and non-homogeneous linear ordinary differential equations will be discussed. Similarly, methods for solving equations in which the coefficients are not constant will be covered as well.
Integral Transforms: In this part of the course, students will review how to express complicated functions into power series and to obtain representation for functions that are defined over an infinite interval and have no particular periodicity. Fourier transforms, as a generalization of Fourier series, and the Laplace transforms are the most used forms of integral transforms. Most-common applications such as, signal processing, solid state physics and solving differential equations of harmonic damped oscillators and RCL circuits will be covered along this unit.
Partial Differential Equations: The solution of differential equations are typically encountered in the engineering and physical sciences where the problem involves mores than one independent variable. This unit will present general solutions of PDEs, uniqueness of the solution to PDEs under given boundary conditions. Application of methods such as, separation of variables, integral transforms and Green’s functions will play a fundamental when solving PDEs.