alan jeffrey

1181 Pages·2012·9.8 MB·English

Alan Jeffrey
University of Newcastle-upon-Tyne
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C O N T E N T S
Preface xv
P A R T O N E REVIEW MATERIAL 1
1
CHAPTER Review of Prerequisites 3
1.1 Real Numbers, Mathematical Induction, and
Mathematical Conventions 4
1.2 Complex Numbers 10
1.3 The Complex Plane 15
1.4 Modulus and Argument Representation of
Complex Numbers 18
1.5 Roots of Complex Numbers 22
1.6 Partial Fractions 27
1.7 Fundamentals of Determinants 31
1.8 Continuity in One or More Variables 35
1.9 Differentiability of Functions of One or
More Variables 38
1.10 Tangent Line and Tangent Plane Approximations
to Functions 40
1.11 Integrals 41
1.12 Taylor and Maclaurin Theorems 43
1.13 Cylindrical and Spherical Polar Coordinates and Change of
Variables in Partial Differentiation 46
1.14 Inverse Functions and the Inverse Function Theorem 49
vii
P A R T T W O VECTORS AND MATRICES 53
2
CHAPTER Vectors and Vector Spaces 55
2.1 Vectors, Geometry, and Algebra 56
2.2 The Dot Product (Scalar Product) 70
2.3 The Cross Product (Vector Product) 77
2.4 Linear Dependence and Independence of Vectors
and Triple Products 82
2.5 n-Vectors and the Vector Space Rn 88
2.6 Linear Independence, Basis, and Dimension 95
2.7 Gram–Schmidt Orthogonalization Process 101
3
CHAPTER Matrices and Systems of Linear Equations 105
3.1 Matrices 106
3.2 Some Problems That Give Rise to Matrices 120
3.3 Determinants 133
3.4 Elementary Row Operations, Elementary Matrices,
and Their Connection with Matrix Multiplication 143
3.5 The Echelon and Row-Reduced Echelon
Forms of a Matrix 147
3.6 Row and Column Spaces and Rank 152
3.7 The Solution of Homogeneous Systems
of Linear Equations 155
3.8 The Solution of Nonhomogeneous Systems
of Linear Equations 158
3.9 The Inverse Matrix 163
3.10 Derivative of a Matrix 171
4
CHAPTER Eigenvalues, Eigenvectors, and Diagonalization 177
4.1 Characteristic Polynomial, Eigenvalues,
and Eigenvectors 178
4.2 Diagonalization of Matrices 196
4.3 Special Matrices with Complex Elements 205
4.4 Quadratic Forms 210
4.5 The Matrix Exponential 215
viii
P A R T T H R E E ORDINARY DIFFERENTIAL EQUATIONS 225
5
CHAPTER First Order Differential Equations 227
5.1 Background to Ordinary Differential Equations 228
5.2 Some Problems Leading to Ordinary
Differential Equations 233
5.3 Direction Fields 240
5.4 Separable Equations 242
5.5 Homogeneous Equations 247
5.6 Exact Equations 250
5.7 Linear First Order Equations 253
5.8 The Bernoulli Equation 259
5.9 The Riccati Equation 262
5.10 Existence and Uniqueness of Solutions 264
6
CHAPTER Second and Higher Order Linear Differential
Equations and Systems 269
6.1 Homogeneous Linear Constant Coefficient Second
Order Equations 270
6.2 Oscillatory Solutions 280
6.3 Homogeneous Linear Higher Order Constant
Coefficient Equations 291
6.4 Undetermined Coefficients: Particular Integrals 302
6.5 Cauchy–Euler Equation 309
6.6 Variation of Parameters and the
Green’s Function 311
6.7 Finding a Second Linearly Independent Solution
from a Known Solution: The Reduction of Order
Method 321
6.8 Reduction to the Standard Form u(cid:4)(cid:4) + f(x)u = 0 324
6.9 Systems of Ordinary Differential Equations:
An Introduction 326
6.10 A Matrix Approach to Linear Systems
of Differential Equations 333
6.11 Nonhomogeneous Systems 338
6.12 Autonomous Systems of Equations 351
ix