## Syllabus & textbook homework for Math 421:01, fall 2005

 The text is Advanced Engineering Mathematics (second edition) by Dennis G. Zill and Michael R. Cullen. It is published by Jones and Barlett, 2000 and has 926+95 [Appendices, Answers and Index] pages (ISBN# 0-763-71065-2). The book is for sale at the Rutgers University bookstore for \$133. It can be bought at Amazon.com with free shipping for less. Math 421 is directed at students in the Mechanical & Aerospace curriculum (650) and designed to help these students prepare for certain senior-level engineering courses. The course is also required for students in Chemical Engineering (155). Here is more information. Time is set aside for two full-period exams. There also will be a final exam. Conceptually the course has three parts. The last time I taught the course I spent too much time on linear algebra and not enough time on PDE's: in particular, I did not even get to 14.1, and I'd like to at least get to this section. Laplace transforms (7 lectures) The definition, simple properties, and applications to the solution of ODE's. Linear algebra (8 lectures) 421 students are assumed to have some background in linear algebra. Therefore part of this is a fast-paced review, combined with an effort to insure that important facts and algorithms are clearly stated and can be used by students in later courses. Fourier series and some boundary value problems for ODE's and PDE's (11 lectures) Fourier series; Fourier sine and cosine series; the wave and heat equations in 1 and 2 space dimensions.

Text section Section title Suggested problems
Review Students who need to review elements of ODE's (as covered in Math 244) should look through Chapter 1 (Introduction to Differential Equations) and at least section 3.1 (Preliminary Theory) of Chapter 2 (Higher-Order Differential Equations).
Goals Students should "remember" what an Ordinary Differential Equation (ODE) is, and remember the Initial Value Problem (IVP) for ODE's. IVP's usually have unique local solutions. Students should also have some familiarity with setting up IVP's for ODE's from physical situations. Constant coefficient second order equations are one very simple model, with solutions involving sin/cos and sinh/cosh mentioned. Students should know alternative descriptions of the solutions using complex numbers and Euler's formula.
4.1 Definition of the Laplace transform 1, 5, 7, 13, 23, 35
4.2 The Inverse Transform and Transform of Derivatives 7, 11, 23, 31, 35, 39
4.3 Translation Theorems 3, 7, 9, 11, 15, 19, 37, 41, 45, 47, 49-54, 67
4.4 Additional Operational Properties 3, 7, 11, 13, 19a, 27, 33, 37, 39
4.5 Dirac Delta Function 1, 3, 9
4.6 Solving Systems of Linear Equations 1, 7, 9, 11
Goals Students should know the definition of the Laplace transform. They should be able to compute simple Laplace transforms "by hand" and should be able to use a table of Laplace transforms to solve initial value problems for constant coefficient ODE's. They should be able to write "rough" data in terms of Dirac and Heaviside functions. They should be able to recognize and use simple results regarding Laplace transforms.
7.6 Vector spaces (refer to this section for definitions as needed).
8.1 Matrix Algebra 17, 23, 29, 35, 37
8.2 Systems of Linear Algebraic Equations 5, 9, 11, 27
8.3 Rank of a Matrix 9, 13, 15, 17, 19
8.4 Determinants 19, 21, 25, 27
8.5 Properties of Determinants 5, 11, 13, 23, 31, 33
8.6 Inverse of a Matrix 7, 25, 27, 31, 51, 53
8.7 Cramer's Rule 1, 9
8.8 The Eigenvalue Problem 5, 13, 19
8.10 Orthogonal Matrices 7, 13, 15
8.12 Diagonalization 5, 13, 15, 23
Goals Students should know the principal definitions of linear algebra and recognize and work with these ideas. Students should know how to compute the rank of a matrix, the determinant of a matrix, and be able to invert and diagonalize simple matrices "by hand".
3.9 Linear Models: Boundary-Value Problems 11, 15
12.1 Orthogonal Functions 3, 7, 17
12.2 Fourier Series 1, 5, 9, 15, 17, 21
Problem 21 is Parseval's Formula
12.3 Fourier Sine and Cosine Series 3, 5, 7, 13, 19, 23, 29, 35
12.4 Complex Fourier Series and Frequency Spectrum 5
12.5 Sturm-Liouville Problem 1
Goals Students should be able to compute the Fourier coefficients of simple functions "by hand". They should know what orthogonality means for functions. Given a function's graph on [0,A] they should be able to graph the sum of the Fourier sine and cosine series on [-A,A]. They should have some qualitative understanding of what partial sums of Fourier series look like compared to the original function, including an idea of the Gibbs phenomenon.
13.1 Separable Partial Differential Equations 3, 11, 15
13.2 Classical Equations and Boundary-Value Problems 1, 3, 5, 9
13.3 Heat Equation 1, 3, 4
13.4 Wave Equation 1, 3, 5, 14, 15, 18
Problem 14 gives d'Alembert's solution to the wave equation.
13.5 Laplace's Equation 5, 7, 15
13.8 Fourier Series in Two Variables 1, 3
14.1 Laplace's Equation in Polar Coordinates 1, 3, 7
15.2 Applications of the Laplace Transform 5, 9
Goals Students should be able to solve simple boundary and initial value problems for the classical PDE's using separation of variables combined with the appropriate trigonometric series. They should also understand some of the qualitative aspects of solutions (smoothness for heat and possible shocks for wave), and simple asymptotics for some BVP's.
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