The mathematics 294 course of Claremont Graduate University introduced the derivation of classical equations of applied mathematics. The equations emphasized were: quasilinear hyperbolic, Laplace, Poisson, Helmholtz, advection-diffusion, and wave equations. Furthermore, most of the equations were examined in various orthogonal curvilinear coordinate systems. For example, the gradient, divergence, curl, and Laplacian operators were studied in Cartesian, cylindrical and spherical coordinates, acting on scalar or vector fields. In addition, methods of solutions on bounded domains using eigenfunction expansions and the associated Sturm-Liouville eigenvalues problems, including Bessel functions and Legendre polynomials were covered.
The class was challenging, but rewarding. I was able to review my basic calculus knowledge and expand my mathematical content knowledge at the same time. The homework assignments were demanding and I frequently sought the help of my classmates, the class tutor, and the professor. The most challenging topic for me was to understand scalar and vector fields, as well as the gradient, divergence, curl and Laplacian operators acting on such fields.
In conclusion, the course increased mathematical content knowledge and made me a better teacher. I was provided with the opportunity to study the classical equations of applied mathematics. For example, we applied the method of characteristics to solve first-order quasi-linear partial differential equations and analyze the formation of shocks and expansion waves in such systems. Prior to the course I knew that differential equations were greatly used within the mathematics field, but was not able to point out several specific examples. In addition, the course increased and sharpened my calculus knowledge, which will make me a better calculus teacher.