حسابان پیشرفته (ویرایش پنجم)

The Fifth Edition of this leading text offers substantial training in vectors and matrices, vector analysis, and partial differential equations. Vectors are introduced at the outset and serve at many points to indicate geometrical and physical significance of mathematical relations. Numerical methods are touched upon at various points, because of their practical value and the insights they give about theory. Vectors and Matrices; Differential Calculus of Functions of Several Variables; Vector Differential Calculus; Integral Calculus of Functions of Several Variables; Vector Integral Calculus; Two-Dimensional Theory; Three-Dimensional Theory and Applications; Infinite Series; Fourier Series and Orthogonal Functions; Functions of a Complex Variable; Ordinary Differential Equations; Partial Differential Equations For all readers interested in 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 A book for an advanced calculus course. Though its relatively small size, the book approaches a wide content. The treatment is simple and objective. Among the topics covered are included several variables differential and integral calculus (vol. 1), vectorial calculus (vol. 1), special functions (vol. 2), sequences and series (vol. 2), Fourier series (vol. 2), complex variables (vol. 2) and ordinary and partial differential equations (vol. 2). Offers training in vectors and matrices, vector analysis, and partial differential equations. This book introduces vectors at the outset and serves at many points to indicate geometrical and physical significance of mathematical relations. It covers numerical methods at various points, because of the insights they give about theory. Wilfred Kaplan. Includes Bibliographical References And Index.