CHAPTER 2
MATHEMATICAL MODELS AND NUMERICAL METHODS
SECTION 2.1 POPULATION MODELS
Section 2.1 introduces the first of the two major classes of mathematical models studied in the textbook, and is a prerequisite to the discussion of equilibrium soluti
5.1 Briefly explain the difference between self-diffusion and interdiffusion.
Answer
Self-diffusion is atomic migration in pure metals-i.e., when all atoms exchanging positions are of the same
type. Interdiffusion is diffusion of atoms of one metal into a
Primary Interatomic Bonds
2.22 (a) Briey cite the main differences among ionic, covalent, and metallic bonding.
(b) State the Pauli exclusion principle.
Solution
(a) The main differences between the various forms of primary bonding are:
lonic-there is ele
CHAPTER 1
FIRST-ORDER DIFFERENTIAL EQUATIONS
SECTION 1.1 DIFFERENTIAL EQUATIONS AND MATHEMATICAL MODELS
The main purpose of Section 1.1 is simply to introduce the basic notation and terminology of differential equations, and to show the student what
APPENDIX A
EXISTENCE AND UNIQUENESS OF SOLUTIONS
In Problems 112 we apply the iterative formula
yn+1 = b + f (t , yn (t ) dt
a
x
to compute successive approximations {yn(x)} to the solution of the initial value problem
y = f (x, y), y(a) = b.
s
CHAPTER 11
POWER SERIES METHODS
SECTION 11.1 INTRODUCTION AND REVIEW OF POWER SERIES
The power series method consists of substituting a series y = cnxn into a given differential equation in order to determine what the coefficients {cn} must be in or
CHAPTER 10
LAPLACE TRANSFORM METHODS
SECTION 10.1 LAPLACE TRANSFORMS AND INVERSE TRANSFORMS
The objectives of this section are especially clearcut. They include familiarity with the definition of the Laplace transform L{f(t)} = F(s) that is given in
CHAPTER 9
NONLINEAR SYSTEMS AND PHENOMENA
SECTION 9.1 STABILITY AND THE PHASE PLANE
1. The only solution of the homogeneous system 2 x - y = 0, x - 3 y = 0 is the origin (0, 0). The only figure among Figs. 9.1.11 through 9.1.18 showing a single crit
CHAPTER 8
MATRIX EXPONENTIAL METHODS
SECTION 8.1
In Problems 18 we first use the eigenvalues and eigenvectors of the coefficient matrix A to find first a fundamental matrix (t) for the homogeneous system x = Ax. Then we apply the formula x(t) = (t)
CHAPTER 7
LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS
SECTION 7.1 FIRST-ORDER SYSTEMS AND APPLICATIONS
1.
Let x1 = x and x2 = x1 = x, so x2 = x = -7 x - 3x + t 2 .
Equivalent system: x1 = x2, 2. x2 = -7x1 - 3x2 + t2
Let x1 = x, x2 = x1 = x, x
CHAPTER 6
EIGENVALUES AND EIGENVECTORS
SECTION 6.1 INTRODUCTION TO EIGENVALUES
In each of Problems 132 we first list the characteristic polynomial p( ) = A - I of the given
matrix A, and then the roots of p ( ) - which are the eigenvalues of A. Al
CHAPTER 5
HIGHER-ORDER LINEAR DIFFERENTIAL EQUATIONS
SECTION 5.1 INTRODUCTION: SECOND-ORDER LINEAR EQUATIONS
In this section the central ideas of the theory of linear differential equations are introduced and illustrated concretely in the context of
CHAPTER 4
VECTOR SPACES
The treatment of vector spaces in this chapter is very concrete. Prior to the final section of the chapter, almost all of the vector spaces appearing in examples and problems are subspaces of Cartesian coordinate spaces of n
CHAPTER 3
LINEAR SYSTEMS AND MATRICES
SECTION 3.1 INTRODUCTION TO LINEAR SYSTEMS
This initial section takes account of the fact that some students remember only hazily the method of elimination for 2 2 and 3 3 systems. Moreover, high school algebr
4.1 The equilibrium fraction of lattice sites that are vacant in silver (Ag) at 700C is 2 106. Calculate the number of
vacancies (per meter cubed) at 700C. Assume a density of 10.35 g/cm3 for Ag.
Solution
This problem is solved using two steps: (1) calcu