This course is an applied mathematics course tailored to the needs of engineering graduate students.  As a result of instruction in this course, the students should be able to skillfully apply the following areas of advanced calculus to various areas of engineering and physical sciences. The course begins with a detailed treatment of complex variable theory, which includes Cauchy’s theorem, classification of isolated singularities, multi-valuedness and branch-cut singularities, Taylor and Laurent series and contour integration methods for evaluation of integrals. Next, we introduce vector spaces and linear operators which then leads to an in-depth treatment of the following three topics: i) Series solution of differential equations and special functions (e.g.: Bessel, Hankel, Legendre, Gamma functions etc.), ii) Sturm-Liouville and boundary value problems, and iii) Generalized Fourier series, eigen-values and eigen-function expansion. These topics then lead to Fourier analysis, and multi-dimensional Fourier series and transforms. A study of partial differential equations, selected topics in higher dimensional calculus, variational calculus and Green’s function theory (if time permitted) will conclude the course. Various examples from engineering and physics will be incorporated in the above topics as appropriate.