There are three chapters in this course. Chapter one covers the Cartesian Tensors, which are extensively used in the courses of Elasticity, Plasticity, Fluid mechanics, and other science subjects. Chapter two includes three parts. The first part introduces the existence and uniqueness theory for a system of 1st order ordinary differential equations (ODE). The second part covers the solution of a system of linear 1st order ODE, which is particular useful for the course of Dynamics. The third part is designed to the solution of linear 2nd order ODE with unknown source functions. We introduce the concepts of Dirac delta function, generalized functions, adjoint operators, Green’s functions and the integral representation of the solution of 2nd order ODE. Chapter 3 also includes three parts. The 1st part introduces the classification of the 2nd order partial differential equations (PDE). The 2nd part introduces the Green’s function and the integral representation of solution of 2nd order linear PDEs. Free space Green’s functions are solved first for infinite domain and then method of images are introduced for solving some PDE with simple finite domains. The 3rd part introduces the eigenvalue problem of self-adjoint boundary value problems of 2nd order PDE, and the full or partial eigenfunction expansion for solving the linear 2nd order BVP or IBVP. Also included in this part are the Maximum-Minimum principle and unique theorems for Laplace/Poisson equation and Heat equation.

This course is aimed to let the graduate students acquire the analytic skills needed in mechanics, electricity and applied science.

The students who take this course for credits should have taken at least one year of engineering mathematics courses in most engineering departments or equivalent courses which contain vector and matrix analysis and differential equations.

1) H. Jeffreys, "Cartesian tensors," 7th ed., Cambridge Univ. Press, 1968. (2) Y. C. Fung, "A first course in continuum mechanics," Prentice-Hall, 1969. (3) G. Birkho and G. C. Rota, "Ordinary Differential Equations," 4th ed. John Wiley & Sons, 1989. (5) F. Brauer J. A. Nohel, "Ordinary Differential Equations," Benjamin Inc., 1967. (6) I. Stakgold, "Green's Functions and Boundary Value Problems," John Wiley & Sons., 1979. (7) M. W. Hirsch and S. Smale, "Differential Equations, Dynamical Systems, and Linear Algebra," Academic Press, 1974. (8) W. E. Williams,“Partial differential equations,” Oxford University Press, 1980.