Advanced Calculus

FrontPage......Page 1
Preface......Page 2
1.Vectors and Matrices......Page 6
1.2 Vectors in Space......Page 13
1.3 Linear Independence: Lines and Planes......Page 18
1.4 Determinants......Page 21
1.5 Simultaneous Linear Equations......Page 25
1.6 Matrices......Page 30
1.7 Addition of Matrices & Scalar Times Matrix......Page 31
1.8 Multiplication of Matrices......Page 33
1.9 Inverse of a Square Matrix......Page 38
1.10 Gaussian Elimination......Page 44
1.11 Eigenvalues of a Square Matrix......Page 47
*1.12 The Transpose......Page 51
*1.13 Orthogonal Matrices......Page 53
1.14 Analytic Geometry and Vectors in n-Dimensional Space......Page 58
*1.15 Axioms for Vn......Page 63
1.16 Linear Mappings......Page 67
*1.17 Subspaces rn Rank of a Matrix......Page 74
*l.l8 Other Vector Spaces......Page 79
2.1 Functions of Several Variables......Page 85
Index......Page 0
2.3 Functional Notation rn Level Curves and Level Surfaces......Page 88
2.4 Limits and Continuity......Page 90
2.5 Partial Derivatives......Page 95
2.6 Total Differential Fundamental Lemma......Page 98
2.7 Differential of Functions of n Variables The Jacobian Matrix......Page 102
2.8 Derivatives and Differentials of Composite Functions......Page 108
2.9 The General Chain Rule......Page 113
2.10 Implicit Functions......Page 117
*2.11 Proof of a Case of the Implicit Function Theorem......Page 124
2.12 Inverse Functions Curvilinear Coordinates......Page 130
2.13 Geometrical Applications......Page 134
2.14 The Directional Derivative......Page 143
2.15 Partial Derivatives of Higher Order......Page 147
2.16 Higher Derivatives of Composite Functions......Page 150
2.18 Higher Derivatives of Implicit Functions......Page 152
2.19 Maxima and Minima of Functions of Several Variables......Page 154
*2.20 Extrema for Functions with Side Conditions LagrangeMultipliers......Page 157
*2.21 Maxima and Minima of Quadratic Forms on the Unit Sphere......Page 166
*2.22 Functional Dependence......Page 167
*2.23 Real Variable Theory rn Theorem on Maximum and Minimum......Page 179
5.8 Line Integrals in Space......Page 8
7.1 Trigonometric Series......Page 9
8.1 Complex Functions......Page 10
10.1 Introduction......Page 11