Math IV – Engineering Mathematics II

There are essentially three separate components of the course. The first component involves a discussion of multiple integrals and vector calculus. This material can best be described as the mathematics needed to study fluids. The course covers the theory of vector-valued functions. Multiple integrals are covered extensively. Emphasis is placed on the transformation of space/coordinates and the role of the Jacobian. The concepts of vector and scalar fields, curl, and divergence are introduced from a very physical point of view, as are line and surface integrals. The three major theorems of vector Calculus – Green’s theorem, Stokes’ theorem, and the Divergence (Gauss’) theorem are covered. A strong emphasis is placed on physical interpretation. This material is highly visual and makes extensive use of computer graphing to illustrate the concepts. The second component of this class involves complex variables. This component covers the basic arithmetic and geometry of the complex number system. Then the calculus of functions of complex numbers is studied, including the Cauchy-Riemann equations and the implications for harmonic functions. Complex exponential, trigonometric, and logarithmic functions are defined and studied. There is a brief treatment of conformal mapping. In addition, basic integral procedures are discussed. The third component of the course covers the remaining essential parts of linear algebra and differential equations. The class works more extensively with matrices, eigenvalues and eigenvectors, matrix functions, and the calculus of matrix functions. The concepts are applied to the solution of the systems of first-order differential equations. The course meets for four hours per week in the second semester.