Introduction to Calculus and Analysis

The new Chapter 1 contains all the fundamental properties of linear differential forms and their integrals. These prepare the reader for the introduction to higher-order exterior differential forms added to Chapter 3. Also found now in Chapter 3 are a new proof of the implicit function theorem by successive approximations and a discus­ sion of numbers of critical points and of indices of vector fields in two dimensions. Extensive additions were made to the fundamental properties of multiple integrals in Chapters 4 and 5. Here one is faced with a familiar difficulty: integrals over a manifold M, defined easily enough by subdividing M into convenient pieces, must be shown to be inde­ pendent of the particular subdivision. This is resolved by the sys­ tematic use of the family of Jordan measurable sets with its finite intersection property and of partitions of unity. In order to minimize topological complications, only manifolds imbedded smoothly into Euclidean space are considered. The notion of "orientation" of a manifold is studied in the detail needed for the discussion of integrals of exterior differential forms and of their additivity properties. On this basis, proofs are given for the divergence theorem and for Stokes's theorem in n dimensions. To the section on Fourier integrals in Chapter 4 there has been added a discussion of Parseval's identity and of multiple Fourier integrals.

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 parallelepipeds in higher dimensions, 190 d. Orientation of parallelepipeds in n-dimensional space, 195 e. Orientation of planes and hyperplanes, 200 f. Change of volume of parallelepipeds in linear transformations, 201.- 2.5 Vector Notions in Analysis.- a. Vector fields, 204 b. Gradient of a scalar, 205 c. Divergence and curl of a vector field, 208 d. Families of vectors. Application to the theory of curves in space and to motion of particles, 211.- 3 Developments and Applications of the Differential Calculus.- 3.1 Implicit Functions.- a. General remarks, 218 b. Geometrical interpretation, 219 c. The implicit function theorem, 221 d. Proof of the implicit function theorem, 225 e. The implicit function theorem for more than two independent variables, 228.- 3.2 Curves and Surfaces in Implicit Form.- a. Plane curves in implicit form, 230 b. Singular points of curves, 236 c. Implicit representation of surfaces, 238.- 3.3 Systems of Functions, Transformations, and Mappings.- a. General remarks, 241 b. Curvilinear coordinates, 246 c. Extension to more than two independent variables, 249 d. Differentiation formulae for the inverse functions, 252 e. Symbolic product of mappings, 257 f. General theorem on the inversion of transformations and of systems of implicit functions. Decomposition into primitive mappings, 261 g. Alternate construction of the inverse mapping by the method of successive approximations, 266 h. Dependent functions, 268 i. Concluding remarks, 275.- 3.4 Applications.- a. Elements of the theory of surfaces, 278 b. Conformal transformation in general, 289.- 3.5 Families of Curves, Families of Surfaces, and Their Envelopes.- a. General remarks, 290 b. Envelopes of one-parameter families of curves, 292 c. Examples, 296 d. Endevelopes of families of surfaces, 303.- 3.6 Alternating Differential Forms.- a. Definition of alternating differential forms, 307 b. Sums and products of differential forms, 310 c. Exterior derivatives of differential forms, 312 d. Exterior differential forms in arbitrary coordinates, 316.- 3.7 Maxima and Minima.- a. Necessary conditions, 325 b. Examples, 327 c. Maxima and minima with subsidiary conditions, 330 d. Proof of the method of undetermined multipliers in the simplest case, 334 e. Generalization of the method of undetermined multipliers, 337 f. Examples, 340.- A.1 Sufficient Conditions for Extreme Values.- A.2 Numbers of Critical Points Related to Indices of a Vector Field.- A.3 Singular Points of Plane Curves.- A.4 Singular Points of Surfaces.- A.5 Connection Between Euler’s and Lagrange’s Representation of the motion of a Fluid.- A.6 Tangential Representation of a Closed Curve and the Isoperimetric Inequality.- 4 Multiple Integrals.- 4.1 Areas in the Plane.- a. Definition of the Jordan measure of area, 367 b. A set that does not have an area, 370 c. Rules for operations with areas, 372.- 4.2 Double Integrals.- a. The double integral as a volume, 374 b. The general analytic concept of the integral, 376 c. Examples, 379 d. Notation. Extensions. Fundamental rules, 381 e. Integral estimates and the mean value theorem, 383.- 4.3 Integrals over Regions in three and more Dimensions.- 4.4 Space Differentiation. Mass and Density.- 4.5 Reduction of the Multiple Integral to Repeated Single Integrals.- a. Integrals over a rectangle, 388 b. Change of order of integration. Differentiation under the integral sign, 390 c. Reduction of double integrals to single integrals for more general regions, 392 d. Extension of the results to regions in several dimensions, 397.- 4.6 Transformation of Multiple Integrals.- a. Transformation of integrals in the plane, 398 b. Regions of more than two dimensions, 403.- 4.7 Improper Multiple Integrals.- a. Improper integrals of functions over bounded sets, 407 b. Proof of the general convergence theorem for improper integrals, 411 c. Integrals over unbounded regions, 414.- 4.8 Geometrical Applications.- a. Elementary calculation of volumes, 417 b. General remarks on the calculation of volumes. Solids of revolution. Volumes in spherical coordinates, 419 c. Area of a curved surface, 421.- 4.9 Physical Applications.- a. Moments and center of mass, 431 b. Moments of inertia, 433 c. The compound pendulum, 436 d. Potential of attracting masses, 438.- 4.10 Multiple Integrals in Curvilinear Coordinates.- a. Resolution of multiple integrals, 445 b. Application to areas swept out by moving curves and volumes swept out by moving surfaces. Guldin’s formula. The polar planimeter, 448.- 4.11 Volumes and Surface Areas in Any Number of Dimensions.- a. Surface areas and surface integrals in more than three dimensions, 453 b. Area and volume of the n-dimensional sphere, 455 c. Generalizations. Parametric Representations, 459.- 4.12 Improper Single Integrals as Functions of a Parameter.- a. Uniform convergence. Continuous dependence on the parameter, 462 b. Integration and differentiation of improper integrals with respect to a parameter, 466 c. Examples, 469 d. Evaluation of Fresnel’s integrals, 473.- 4.13 The Fourier Integral.- a. Introduction, 476 b. Examples, 479 c. Proof of Fourier’s integral theorem, 481 d. Rate of convergence in Fourier’s integral theorem, 485 e. Parseval’s identity for Fourier transforms, 488 f. The Fourier transformation for functions of several variables, 490.- 4.14 The Eulerian Integrals (Gamma Function).- a. Definition and functional equation, 497 b. Convex functions. Proof of Bohr and Mollerup’s theorem, 499 c. The infinite products for the gamma function, 503 d. The nextensio theorem, 507 e. The beta function, 508 f. Differentiation and integration of fractional order. Abel’s integral equation, 511.- Appendix: Detailed Analysis of the Process of Integration.- A.1 Area.- a. Subdivisions of the plane and the corresponding inner and outer areas, 515 b. Jordan-measurable sets and their areas, 517 c. Basic properties of areas, 519.- A.2 Integrals of Functions of Several Variables.- a. Definition of the integral of a function f(x, y), 524 b. Integrability of continuous functions and integrals over sets, 526 c. Basic rules for multiple integrals, 528 d. Reduction of multiple integrals to repeated single integrals, 531.- A.3 Transformation of Areas and Integrals.- a. Mappings of sets, 534 b. Trans formation of multiple integrals, 539.- A.4 Note on the Definition of the Area of a Curved Surface.- 5 Relations Between Surface and Volume Integrals.- 5.1 Connection Between Line Integrals and Double Integrals in the Plane (The Integral Theorems of Gauss, Stokes, and Green).- 5.2 Vector Form of the Divergence Theorem. Stokes’s Theorem.- 5.3 Formula for Integration by Parts in Two Dimensions. Green’s Theorem.- 5.4 The Divergence Theorem Applied to the Transformation of Double Integrals.- a. The case of 1–1 mappings, 558 b. Transformation of integrals and degree of mapping, 561.- 5.5 Area Differentiation. Transformation of ?u to Polar Coordinates.- 5.6 Interpretation of the Formulae of Gauss and Stokes by Two-Dimensional Flows.- 5.7 Orientation of Surfaces.- a. Orientation of two-dimensional surfaces in three-space, 575 b. Orientation of curves on oriented surfaces, 587.- 5.8 Integrals of Differential Forms and of Scalars over Surfaces.- a. Double integrals over oriented plane regions, 589 b. Surface integrals of second-order differential forms, 592 c. Relation between integrals of differential forms over oriented surfaces to integrals of scalars over unoriented surfaces, 594.- 5.9 Gauss’s and Green’s Theorems in Space.- a. Gauss’s theorem, 597 b. Application of Gauss’s theorem to fluid flow, 602 c. Gauss’s theorem applied to space forces and surface forces, 605 d. Integration by parts and Green’s theorem in three dimensions, 607 e. Application of Green’s theorem to the transformation of ?U to spherical coordinates, 608.- 5.10 Stokes’s Theorem in Space.- a. Statement and proof of the theorem, 611 b. Interpretation of Stokes’s theorem, 615.- 5.11 Integral Identities in Higher Dimensions.- Appendix: General Theory of Surfaces and of Surface Integals.- A.1 Surfaces and Surface Integrals in Three dimensions.- a. Elementary surfaces, 624 b. Integral of a function over an elementary surface, 627 c. Oriented elementary surfaces, 629 d. Simple surfaces, 631 e. Partitions of unity and integrals over simple surfaces, 634.- A.2 The Divergence Theorem.- a. Statement of the theorem and its invariance, 637 b. Proof of the theorem, 639.- A.3 Stokes’s Theorem.- A.4 Surfaces and Surface Integrals in Euclidean Spaces of Higher Dimensions.- a. Elementary surfaces, 645 b. Integral of a differential form over an oriented elementary surface, 647 c. Simple m-dimensional surfaces, 648.- A.5 Integrals over Simple Surfaces, Gauss’s Divergence Theorem, and the General Stokes Formula in Higher Dimensions.- 6 Differential Equations.- 6.1 The Differential Equations for the Motion of a Particle in Three Dimensions.- a. The equations of motion, 654 b. The principle of conservation of energy, 656 c. Equilibrium. Stability, 659 d. Small oscillations about a position of equilibrium, 661 e. Planetary motion, 665 f. Boundary value problems. The loaded cable and the loaded beam, 672.- 6.2 The General Linear Differential Equation of the First Order.- a. Separation of variables, 678 b. The linear first-order equation, 680.- 6.3 Linear Differential Equations of Higher Order.- a. Principle of superposition. General solutions, 683 b. Homogeneous differential equations of the second second order, 688 c. The nonhomogeneous differential equations. Method of variation of parameters, 691.- 6.4 General Differential Equations of the First Order.- a. Geometrical interpretation, 697 b. The differential equation of a family of curves. Singular solutions. Orthogonal trajectories, 699 c. Theorem of the existence and uniqueness of the solution, 702.- 6.5 Systems of Differential Equations and Differential Equations of Higher Order.- 6.6 Integration by the Method of Undermined Coefficients.- 6.7 The Potential of Attracting Charges and Laplace’s Equation.- a. Potentials of mass distributions, 713 b. The differential equation of the potential, 718 c. Uniform double layers, 719 d. The mean value theorem, 722 e. Boundary value problem for the circle. Poisson’s integral, 724.- 6.8 Further Examples of Partial Differential Equations from Mathematical Physics.- a. The wave equation in one dimension, 727 b. The wave equation in three-dimensional space, 728 c. Maxwell’s equations in free space, 731.- 7 Calculus of Variations.- 7.1 Functions and Their Extrema.- 7.2 Necessary conditions for Extreme Values of a Functional.- a. Vanishing of the first variation, 741 b. Deduction of Euler’s differential equation, 743 c. Proofs of the fundamental lemmas, 747 d. Solution of Euler’s differential equation in special cases. Examples, 748 e. Identical vanishing of Euler’s expression, 752.- 7.3 Generalizations.- a. Integrals with more than one argument function, 753 b. Examples, 755 c. Hamilton’s principle. Lagrange’s equations, 757 d. Integrals involving higher derivatives, 759 e. Several independent variables, 760.- 7.4 Problems Involving Subsidiary Conditions. Lagrange Multipliers.- a. Ordinary subsidiary conditions, 762 b. Other types of subsidiary conditions, 765.- 8 Functions of a Complex Variable.- 8.1 Complex Functions Represented by Power Series.- a. Limits and infinite series with complex terms, 769 b. Power series, 772 c. Differentiation and integration of power series, 773 d. Examples of power series, 776.- 8.2 Foundations of the General Theory of Functions of a Complex Variable.- a. The postulate of differentiability, 778 b. The simplest operations of the differential calculus, 782 c. Conformal transformation. Inverse functions, 785.- 8.3 The Integration of Analytic Functions.- a. Definition of the integral, 787 b. Cauchy’s theorem, 789 c. Applications. The logarithm, the exponential function, and the general power function, 792.- 8.4 Cauchy’s Formula and Its Applications.- a. Cauchy’s f...