MATH 231 (Calculus III)

• ABOUT THE COURSE

  Since the concept of a vector has been greatly generalized in mathematics, a vectorial treatment of mechanics, hydrodynamics, and electrodynamics is now practically standard procedure. The use of vectors not only simplifies and condenses the exposition but also makes mathematical and physical concepts more tangible and easy to grasp. The dot and cross product of Gibbs, so intimately involved in all questions of perpendicularity and parallelism, enable one to write the equations of lines and planes at will and to solve all distance problems in the most natural manner. Even the simple distributive law, gives one the power to use the properties of similar triangles without figures and virtually in the dark. Vector analysis has also breached the walls of the calculus and now puts in an appearance in every modern textbook in that subject. The gradient, divergence, and rotation, in their capacity of invariants, form the natural language of spatial science. Avoidance of them is futile, and using long scalar notations every time they occur is wasteful of time and effort.

MATH 335 (ODEs)

• ABOUT THE COURSE

  Formulation of fundamental natural laws and technological problems in the form of rigorous mathematical models is given prevalently in terms of differential equations (equations that involve functions of a single variable as well as derivatives of these functions). Therefore, differential equations is a most important discipline in mathematical education. Several questions naturally arise.
  
  Just what is a differential equation and what does it signify?
  Where and how do differential equations originate and of what use are they?
  Confronted with a differential equation, what does one do with it, how does one do it, and what are the results of such activity?
  These questions indicate three major aspects of the subject: theory, method, and application. The purpose of this course is to introduce audience to the basic aspects of the subject and at the same time give a brief survey of the three aspects just mentioned. In this course, we shall find answers to the general questions raised above, answers which will become more and more meaningful as we proceed with the study of differential equations. This course provides audience with an easy to follow and comprehensive introduction to differential equations and is intended to serve three purposes:
  
  [i] to provide an accessible introduction to the world of differential equations for students who do nott intend to specialize in this area;
  [ii] to provide a prerequisite course for the more specialized third and fourth year courses in ordinary differential equations, partial differential equations, and dynamical systems,
  [iii]to provide an introduction to the discipline of Applied Mathematics, namely, the formulation and analysis of mathematical models of real-world phenomena. Since many models are based on differential equations, an introductory course in DEs provides a natural vehicle for this purpose.
  
  

MATH 254 (Introduction to Applied Linear Algebra)

MATH 438 (Partial Differential Equations)

• ABOUT THE COURSE

  The study of partial differential equations (PDEs) is a fundamental area of mathematics which links important strands of pure mathematics to applied and computational mathematics. Indeed PDEs are ubiquitous in almost all of the applications of mathematics where they provide a natural mathematical description of phenomena in the physical, natural and social sciences. Partial differential equations and their solutions exhibit rich and complex structures.
  
  This course is focussed on some of the most exciting and promising mathematical ideas in these fields, and those branches of PDE theory that provide a source of physically relevant and mathematically hard problems to stimulate future developments.
  
  In this course, there are will be numerous examples and the emphasis is on applications to almost all areas of science and engineering. There is truly something for everyone here.
  
  This course provides a comprehensive and systematic treatment of linear and nonlinear partial differential equations and their varied applications. Building upon my previous teaching experience, this course contains updated modern examples and applications from areas of fluid dynamics, gas dynamics, plasma physics, nonlinear dynamics, quantum mechanics, nonlinear optics, acoustics, and wave propagation. Methods and properties of solutions will be presented, along with their physical significance, making the course more useful for a diverse readership.