# Mathematics

*Professors Goloubeva and Williams*

Mathematics is an analytic tool used in all of the science and engineering courses. At the same time, by its very nature, mathematics is an abstract science. Mathematics at Webb is presented from the applied viewpoint, drawing upon the physical world for its motivation, developing the abstract concepts to refine the physical ideas, and finally applying those abstractions to a better understanding of the phenomena of nature. Many Webb students go on to graduate work involving higher mathematics, and it is a strong objective of the mathematics program to prepare them well for this work.

__Freshman Year__

**MATHEMATICS I - CALCULUS I**

This is an introductory course whose main purpose is to fill in the background of students who have already had an exposure to calculus in high school. The topics covered include limits, continuity, a review of trigonometry, derivatives of functions of one variable, the Riemann Integral and applications of derivatives and integrals to simple physical problems involving related rates, optimization, volume and center of mass. The mean value theorem is covered and extended to series approximations and applied to limits via L'Hopital's rule. The course also covers the elementary transcendental functions of exponentials, logarithms, inverse trigonometric functions, hyperbolic functions and their inverses and concludes with a quick review of the basic techniques of integration such as integration by parts and partial fraction decomposition. The pace is brisk.

The class meets four hours per week in the first semester.

** MATHEMATICS II - CALCULUS II**

This course contains most of the calculus of functions of several variables and includes partial derivatives, multiple integrals and gradients. Vectors are discussed along with the geometry of three-space. There is considerable emphasis of visualization and graphic representation of figures by the computer. The dot product is introduced in such a way as to facilitate its extension to higher dimensional space and ultimately this becomes the inner product in the function space of continuous functions over an interval. This function space approach is used to develop the generalized Fourier series. The trigonometric Fourier series is studied as a special case of the generalized Fourier series. The course also includes an introduction to linear algebra, which covers the rudiments of matrix algebra. Determinants are also introduced here. The course concludes with a unified discussion of real infinite series. The class meets four hours per week in the second semester.

__Sophomore Year__

** MATHEMATICS III - DIFFERENTIAL EQUATIONS**

This class is a basic course in differential equations. It covers the fundamental theorems that guarantee when differential equations and initial value problems, which involve first order linear and non-linear differential equations, have solutions. Linear first order ordinary differential equations with constant coefficients are first studied and general principles are developed which are applicable to all linear ordinary differential equations. This includes integrating factors, variation of parameter techniques and applications to RLC electrical circuits. This study of the linear ordinary differential equations also includes Taylor's series and series solution techniques and problems with singularities. Laplace transform techniques are studied for initial value problems. The course concludes with a review of Fourier series, which are then applied to the basic three forms of initial value problems involving partial differential equations. The class meets three hours per week in the first semester.

** MATHEMATICS IV - ADVANCED ENGINEERING MATHEMATICS**

There are essentially three separate components to the course. The first component involves vector calculus and lasts for about 7 weeks. This material can best be described as the mathematics that is needed to study fluids. The course covers vector functions, gradients and vector and scalar field. Multiple integrals are covered more extensively than in Mathematics II and emphasis is placed on transformation of space/coordinates and the role of the Jacobian. The curl and divergence are introduced from a very physical point of view and their meaning in fluids is covered. Finally the three major theorems of vector calculus, Green's theorem, Stoke's theorem and the Divergence (Gauss') theorem are covered. A strong emphasis is placed on physical interpretation. This material is highly visual and extensive use is made of Maple to illustrate the concepts.

The second component of this class involves complex variables and lasts for about 7 weeks. 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 and there is a brief treatment of conformal mapping. In addition the standard Cauchy integral theorems are treated and these are related to Taylor's and Laurent series.

The third component of this course lasts about 10 class sessions and covers the remaining essential parts of linear algebra to which the students have not previously been exposed. The main focus is on finding and understanding eigenvalues and eigenvectors and what these mean for reduction of matrices into simpler equivalent forms. The class covers the Cayley-Hamilton theorem on the use of the characteristic polynomial to find inverses of matrices. We also cover the reduction of matrices to Jordan canonical forms. Part of the class is devoted to the calculus of matrices, which involves finding functions of matrices and using these functions to solve large systems of differential equations.

The course meets four hours per week in the second semester.

### Junior Year

**PROBABILITY AND STATISTICS**

This course begins with an introduction to probability theory, including set theoretic and combinatorial concepts. This is followed by treatments of discrete random variables and distributions and continuous random variables. Generating functions are discussed at some length. Particular emphasis is placed on the Rayleigh and Weibull distributions, which are applied subsequently in the Ship Dynamics course as models of wave spectra and are also encountered as models of the manufacturing process. The latter third of this course addresses the application of statistical methods to engineering experimentation, beginning with an introduction to estimation and hypothesis testing and culminating with an overview of experiment design. The course meets four hours per week in the first semester.