# Mathematics

*Professors Goloubeva and Williams*

Mathematics is an analytical tool used in all science and engineering courses. At the same time, by its very nature, mathematics is an abstract science. Mathematics at Webb is presented with the focus on applied mathematics, a branch of mathematics which is drawing on the physical world for its motivation, developing abstract concepts to refine the physical ideas, and finally applying those abstractions to mathematical modeling and 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 goals are to fill in the background of students who have already had an exposure to calculus in high school, to deepen their understanding of the material, and to develop their ability for abstract reasoning and mathematical modeling. The course starts with a discussion of vectors in the plane and in space, and basic vector operations, including dot and cross products. The course continues with a review of the real number system and inequalities, algebra of complex numbers, and the theory of elementary functions such as exponentials, logarithms, trigonometric functions, inverse trigonometric functions, hyperbolic functions and their inverses. The topics covered include limits, continuity, derivatives of functions of one variable, application of derivatives to curve sketching and to simple real life problems involving related rates, and optimization. The mean value theorem is covered. Linear and Taylor polynomial approximations are discussed, and applied to limits via L’Hopital’s rule. The course includes a discussion of basic numerical methods such as method of bisections and Newton’s method. The course concludes with a brief discussion of integration. To develop students’ ability for abstract reasoning and to reach a deeper understanding of the material, the discussion often includes proofs.The pace is brisk.

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

** MATHEMATICS II - CALCULUS II**

The course starts with a discussion of integration, integration techniques and applications of integrals. The topics of discussion include the Riemann Integral, the review of the basic techniques of integration such as substitution, integration by parts, partial fraction decomposition and trigonometric techniques of integration. The course covers applications of definite integrals to simple problems involving area between curves, arc length, volume, projectile motion, work and center of mass. The course continues with a discussion of the theory of parametric equations, plane curves, and polar coordinates. The concepts of calculus are extended to curves described by parametric equations and polar coordinates. In this course the geometry of three-space is covered more extensively than in Mathematics I. Cylindrical and spherical coordinates are introduced. Emphasis is placed on visualization and graphical representation of surfaces in space.

This course contains most of the calculus of functions of several variables and includes concepts of limits and continuity of functions of several variables, partial derivatives, tangent planes and linear approximations, gradients, differentials and directional derivatives. In this course we introduce the mathematical basis for finding the maximum or minimum of functions of several variables. Optimization problems for functions of several variables are introduced. The discussion includes constrained optimization and the method of Lagrange multipliers. The course also includes a brief 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 starts with classification of differential equations. It continues with the discussion of ordinary differential equations (ODEs), starting with first order ordinary differential equations. The concepts of direction fields, boundary, and initial value problems are introduced. Several methods of solution of first-order ODEs are considered. It is emphasized that each method is applicable to a certain subclass of first-order equations. The topics covered include methods of solutions of linear equations, separable equations, homogeneous and exact equations, Bernoulli and autonomous equations. The main methods under discussion are the integration of factors, variation of parameters, and separation of variables. The idea of approximating a solution by numerical computation is introduced in the discussion of Euler’s method.

The course continues with a discussion of the general theory and methods of solution of second and nth order ordinary differential equations. Conditions for the existence and uniqueness of the solution are analyzed. Substantial attention is given to methods of solution of second-order differential equations with constant coefficients. Methods under discussion are the reduction of order, undetermined coefficients, the variation of

parameters, series solutions, and the Laplace transform.

The course concludes with a discussion of partial differential equations, Fourier series, and separation of variables as a method for solving partial differential equations. Throughout the course, applications of differential equations to simple physical problems are thoroughly discussed.The class meets three hours per week in the first semester.

** MATHEMATICS IV - ADVANCED ENGINEERING MATHEMATICS**

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 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, Stoke’s 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 Maple 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, standard integral procedures are discussed.

The third component of this course covers the remaining essential parts of linear algebra and differential equations. The class works more extensively with matrices, matrix functions, and the calculus of matrix functions. Then it discusses methods of solution of systems of first-order linear equations, eigenvalues and eigenvectors, and methods of solution of 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.