## Syllabus & textbook homework for Math 421:02, spring 2004

 The text is Advanced Engineering Mathematics (fifth edition) by Peter V. O'Neil. It is published by Brooks/Cole, 2003 and has 1236+82[Answers]+9[Index] pages (ISBN# 0-534-40077-9). Each entry in the table below represents approximately 1 lecture. 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. Here is more information. Time is set aside for two full-period exams. There also will be a cumulative final exam. Conceptually the course has three parts. Laplace transforms The definition and applications to the solution of ODE's. Linear algebra 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 classical boundary value problemsFourier sine and cosine series; the wave and heat equations in 1 and 2 space dimensions.

Section(s)
of text
Section title(s) Suggested problems
3.1 Def'n of Laplace transform and basic properties 3, 5, 7, 13, 15, 17, 27, 29.
3.2 Solution of initial value problems using the Laplace transform 3, 5, 7, 9.
3.3 Shifting theorems and the Heaviside function 1, 3, 7, 9, 13, 15, 17, 19, 21, 25, 31, 37, 47, 49.
3.4 Convolution 1, 3, 7, 9, 11, 13, 17, 19, 23.
3.5 Unit impulses and the Dirac delta function 1, 3, 7, 9, 13, 19.
3.6 Laplace transform solution of systems 1, 3, 5, 7, 9, 11, 15.
3.7 Differential equations with polynomial coefficients 1, 3, 7, 11.
5.4
5.5
The vector space R^n
Linear independence, spanning sets, and dimension in R^n
1, 7, 9, 14, 15, 19, 26.
1, 2, 5, 17, 19, 21
6.1 Matrices 10, 21, 27, 28, 29
6.3
6.4
Row echelon form of a matrix
Row & column spaces of a matrix and rank
1, 3, 9.
1, 3, 9.
6.5 Solution of a homogeneous system of linear equations 1, 5, 13.
6.7 Nonhomogeneous systems of linear equations 1, 5, 9.
6.9 Matrix inverses 1, 7, 13.
7.1
7.2
7.3
7.4
Permutations
Definition of determinant
Properties of determinant
Evaluation of determinant by elementary row & column ops.

1, 7
7.5
7.6
Cofactor expansions
Determinants of triangular matrices
5, 13, 19 a, b.
1
7.7
7.8
Determinant formula for matrix inverse
Cramer's Rule
1, 7.
1, 5, 9.
8.1 Eigenvalues and eigenvectors 1 a, b, 7 a, b, 19 a, b.
8.2 Diagonalization of matrices 1, 7
13.1
13.2
Why Fourier series?
The Fourier series of a function
1
1, 5, 7, 9, 11.
13.4 Fourier cosine and sine series 1, 3, 5, 7.
13.5 Integration and differentiation of Fourier series 3, 5.
16.1
16.2
The wave equation and initial and boundary conditions
Fourier series solution of the wave equation
1, 3.
1, 3, 5, 9.
16.2
16.4

Fourier series and d'Alembert solutions of the wave equation

1, 3, 5, 11, 13, 17.
16.7 Vibrations of a rectangular membrane
Not covered in this class!
1, 3.
17.1
17.2
The heat equation and initial and boundary conditions
Fourier series solution of the heat equation
1, 2, 3.
1, 3, 7, 9, 11.
17.2
17.5

Heat conduction on a rectangular plate
Not covered in this class!
7, 13, 15.
3