Introduction to Calculus and Analysis: Volume II

by Richard Courant

1 Functions of Several Variables and Their Derivatives.- 1.1 Points and Points Sets in the Plane and in Space.- a. Sequences of points. Convergence, 1 b. Sets of points in the plane, 3 c. The boundary of a set. Closed and open sets, 6 d. Closure as set of limit points, 9 e. Points and sets of points in space, 9.- 1.2 Functions of Several Independent Variables.- a. Functions and their domains, 11 b. The simplest types of functions, 12 c. Geometrical representation of functions, 13.- 1.3 Continuity.- a. Definition, 17 b. The concept of limit of a function of several variables, 19 c. The order to which a function vanishes, 22.- 1.4 The Partial Derivatives of a Function.- a. Definition. Geometrical representation, 26 b. Examples, 32 c. Continuity and the existence of partial derivatives, 34 d. Change of the order of differentiation, 36.- 1.5 The Differential of a Function and Its Geometrical Meaning.- a. The concept of differentiability, 40 b. Directional derivatives, 43 c. Geometric interpretation of differentiability, The tangent plane, 46 d. The total differential of a function, 49 e. Application to the calculus of errors, 52.- 1.6 Functions of Functions (Compound Functions) and the Introduction of New Independent Variables.- a. Compound functions. The chain rule, 53 b. Examples, 59 c. Change of independent variables, 60.- 1.7 The Mean Value Theorem and Taylor's Theorem for Functions of Several Variables.- a. Preliminary remarks about approximation by polynomials, 64 b. The mean value theorem, 66 c. Taylor's theorem for several independent variables, 68.- 1.8 Integrals of a Function Depending on a Parameter.- a. Examples and definitions, 71 b. Continuity and differentiability of an integral with respect to the parameter, 74 c. Interchange of integrations. Smoothing of functions, 80.- 1.9 Differentials and Line Integrals.- a. Linear differential forms, 82 b. Line integrals of linear differential forms, 85 c. Dependence of line integrals on endpoints, 92.- 1.10 The Fundamental Theorem on Integrability of Linear Differential Forms.- a. Integration of total differentials, 95 b. Necessary conditions for line integrals to depend only on the end points, 96 c. Insufficiency of the integrability conditions, 98 d. Simply connected sets, 102 e. The fundamental theorem, 104.- A.1. The Principle of the Point of Accumulation in Several Dimensions and Its Applications.- a. The principle of the point of accumulation, 107 b. Cauchy's convergence test. Compactness, 108 c. The Heine-Borel covering theorem, 109 d. An application of the Heine-Borel theorem to closed sets contains in open sets, 110.- A.2. Basic Properties of Continuous Functions.- A.3. Basic Notions of the Theory of Point Sets.- a. Sets and sub-sets, 113 b. Union and intersection of sets, 115 c. Applications to sets of points in the plane, 117.- A.4. Homogeneous functions.- 2 Vectors, Matrices, Linear Transformations.- 2.1 Operations with Vectors.- a. Definition of vectors, 122 b. Geometric representation of vectors, 124 c. Length of vectors. Angles between directions, 127 d. Scalar products of vectors, 131 e. Equation of hyperplanes in vector form, 133 f. Linear dependence of vectors and systems of linear equations, 136.- 2.2 Matrices and Linear Transformations.- a. Change of base. Linear spaces, 143 b. Matrices, 146 c. Operations with matrices, 150 d. Square matrices. The reciprocal of a matrix. Orthogonal matrices. 153.- 2.3 Determinants.- a. Determinants of second and third order, 159 b. Linear and multilinear forms of vectors, 163 c. Alternating multilinear forms. Definition of determinants, 166 d. Principal properties of determinants, 171 e. Application of determinants to systems of linear equations. 175.- 2.4 Geometrical Interpretation of Determinants.- a. Vector products and volumes of parallelepipeds in three-dimensional space, 180 b. Expansion of a determinant with respect to a column. Vector products in higher dimensions, 187 c. Areas of parallelograms and volumes of p

979 pages, Paperback

First published October 21, 2011

About the author

Richard Courant (January 8, 1888 – January 27, 1972) was a German American mathematician. He is best known by the general public for the book What is Mathematics?, co-written with Herbert Robbins.