Calculus V


1. Shot description: The course will discuss partial differential equations, emphasizing those, which arise in science and engineering. The methods used will be elementary and series expansions in terms of trigonometric functions and more general functions of a similar nature. Separation of variables will be used to study various boundary value problems, reducing partial differential equations often to ordinary differential equations. This, in turn, will require the extensive study of Fourier series and special functions such as Bessel functions, Legendre polynomials, etc. This course is also concerned with some ideas of Fourier analysis and Fourier transforms, applications of Fourier’s ideas to the solution of partial differential equations, and the theory of functions of complex variables and its applications in various branches of science and engineering. Topics included are: power series solutions of differential equations, vector and matrices (including vectors space, subspace, linear combination and linear independence, basis, rank, linear transformations, eigenvalues and eigenvectors, diagonalization), Fourier analysis and Fourier transforms, partial differential equations and boundary value problems, analytic functions, CauchyRiemann conditions, elementary functions, simple mappings, complex integrations, Taylor's and Laurent's expansions; the calculus of residues and its applications in computing integrals, construction of conformal mappings between domains, harmonic function and the Dirichlet problem, 2. Course Objective: On successful completion a student should understand: (1) The nature and classification of linear, semilinear, quasilinear homogeneous or inhomogeneous, elliptic, parabolic or hyperbolic boundary value problems; (2) The conservation form of some PDEs and their derivation from conservation principles; (3) The method of characteristics applied to first order PDE's in two variables, including shocks and expansion fans; (4) The method of characteristics applied to linear hyperbolic secondorder PDEs; (5) The method of separation of variables, using orthogonality of eigenfunctions, for solving the heat equation, Laplace's equation and the wave equation. We also introduce students to the basic elements of complex analysis, with particular emphasis on Cauchy’s Theorem and the calculus of residues. On successful completion of the course students will: (1) Develop several kinds of transforms based on Fourier series and integrals, (2) Apply Fourier's ideas to the solution of partial differential equations and boundary value problems, (3) Be familiar with differentiation and integration of functions of complex variables. (4) Have acquired knowledge of Cauchy's Theorem and its applications to evaluating integrals and series. (5) Be able to compute Taylor and Laurent expansions, and to calculate residues. (6) Construct mappings with certain properties between different kinds of sets, and demonstrate how these techniques are used in solving a variety of problems. (7) Demonstrate knowledge of: linear transformations and their representation by matrices. (8) Know the basic algebraic processes for matrices. (9) To know the transpose of a matrix; symmetric, skewsymmetric, and orthogonal matrices. (10) Reduce a matrix to rowechelon form, linear dependence and rank. (11) Know the inverse of a nonsingular square matrix. Determinants. Cramer's rule. (12) Compute eigenvalues and eigenvectors of a square matrix. (13) Reduce orthogonal matrices to diagonal form. 3. Homework is an important part of learning mathematics and will be assigned daily. Every assigned problem should be tried and the answer checked. It is permissible to discuss problems with other students or relatives. It is not permissible to copy another student's work. Do your best to think through the problems and understand why things work the way they do. Homework should take anywhere from 30 minutes to an hour, depending on the type of assignment. Homework will be issued in class and submitted one week later. On test days all homework associated with that test will be collected and graded. Homework should be corrected and added, as the material is better understood. In class, students follow directions: (1) Be in your seat when the bell rings; (2) Do not work on other subjects; (3) No eating, drinking, or chewing; (4) Most of the time the consequences for not following rules will be remaining after class. If it becomes more of a problem, the student will be asked to stay a short time after school for a discussion. 4. Textbook: (1) P. O’Neill, Advanced Engineering Mathematics, 4^{th} ed, Brooks/Cole (1999). (2) Courant, Richard and John, Fritz. Introduction to Calculus and Analysis, New York: SpringerVerlag. (3) J. Stewart, Single Variable Calculus, 4^{th} ed, Brooks/Cole (1999). 