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Chapter 1 Functions of Several Variables and Their Derivatives 1.1 Points and Points Sets in the Plane and in Space a. Sequences of points. Conver- gence, 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 func- tions, 12 c. Geometrical representa- tion of functions, 13 1.3 Continuity a. Definition, 17 b. The concept of limit of a function of several vari- ables, 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 differentia- bility, 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 (Com- pound Functions) and the Introduction of New In- dependent 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 in- dependent variables, 68 1.8 Integrals of a Function Depend- ing 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 ¡­¡­ Chapter 2 Vectors, Matrices, Linear Transformations Chapter 3 Developments and Applications of the Differential Calculus Chapter 4 Multiple Integrals Chapter 5 Relations Between Surface and Volume Integrals Chapter 6 Differential Equations Chapter 7 Calculus of Variations Chapter 8 Functions of a Complex Variable List of Biographical Dates Index