Introduction to Calculus and Analysis, Vol. 2

Title page......Page 1
Preface......Page 3
Contents......Page 5
a. Sequences of points. Convergence......Page 25
b. Sets of points in the plane......Page 27
c. The boundary of a set. Closed and open sets......Page 30
e. Points and sets of points in space......Page 33
a. Functions and their domains......Page 35
b. The simplest s of functions......Page 36
c. Geometrical representation of functions......Page 37
a. Definition......Page 41
b. The concept of limit of a function of several variables......Page 43
c. The order to which a function vanishes......Page 46
a. Definition. Geometrical representation......Page 50
b. Examples......Page 56
c. Continuity and the existence of partial derivatives......Page 58
d. Change of the order of differentiation......Page 60
a. The concept of differentiability......Page 64
b. Directional derivatives......Page 67
c. Geometric interpretation of differentiability. The tangent plane......Page 70
d. The total differential of a function......Page 73
e. Application to the calculus of errors......Page 76
a. Compound functions. The chain rule......Page 77
b. Examples......Page 83
c. Change of independent variables......Page 84
a. Preliminary remarks about approximation by polynomials......Page 88
b. The mean value theorem......Page 90
c. Taylor's theorem for several independent variables......Page 92
a. Examples and definitions......Page 95
b. Continuity and differentiability of an integral with respect to the parameter......Page 98
c. Interchange of integrations. Smoothing of functions......Page 104
a. Linear differential forms......Page 106
b. Line integrals of linear differential forms......Page 109
c. Dependence of line integrals on endpoints......Page 116
a. Integration of total differentials......Page 119
b. Necessary conditions for line integrals to depend only on the end points......Page 120
c. Insufficiency of the integrability conditions......Page 122
d. Simply connected sets......Page 126
e. The fundamental theorem......Page 128
Appendix......Page 130
a. The principle of the point of accumulation......Page 131
b. Cauchy's convergence test. Compactness......Page 132
c. The Heine-Borel covering theorem......Page 133
d. An application of the Heine-Borel theorem to closed sets contains in open sets......Page 134
A.2. Basic Properties of Continuous Functions......Page 136
a. Sets and sub-sets......Page 137
b. Union and intersection of sets......Page 139
c. Applications to sets of points in the plane......Page 142
A.4. Homogeneous functions......Page 143
a. Definition of vectors......Page 146
b. Geometric representation of vectors......Page 148
c. Length of vectors. Angles between directions......Page 151
d. Scalar products of vectors......Page 155
e. Equation of p-yperplanes in vector form......Page 157
f. Linear dependence of vectors and systems of linear equations......Page 160
a. Change of base. Linear spaces......Page 167
b. Matrices......Page 170
c. Operations with matrices......Page 174
d. Square matrices. The reciprocal of a matrix. Orthogonal matrices......Page 177
a. Determinants of second and third order......Page 183
b. Linear and multilinear forms of vectors......Page 187
c. Alternating multilinear forms. Definition of determinants......Page 190
d. Principal properties of determinants......Page 195
e. Applioation of determinants to systems of linear equations......Page 199
a. Vector products and volumes of parallelepipeds in three-dimensional space......Page 204
b. Expansion of a determinant with respect to a column. Vector products in higher dimensions......Page 211
c. Areas of parallelograms higher dimensions......Page 214
d. Orientation of parallelepipeds in n-dimensional space......Page 219
e. Orientation of planes and hyperplanes......Page 224
f. Change of volume of parallelepipeds in linear transformations......Page 225
a. Vector fields......Page 228
b. Gradient of a scalar......Page 229
c. Divergence and curl of a vector field......Page 232
d. Families of vectors. Application to the theory of curves in space and to motion of particles......Page 235
a. General remarks......Page 242
b. Geometrical interpretation......Page 243
c. The implicit function theorem......Page 245
d. Proof of the implicit function theorem......Page 249
e. The implicit function theorem for more than two independent variables......Page 252
a. Plane curves in implicit form......Page 254
b. Singular points of curves......Page 260
c. Extension to more than two independent variables......Page 273
a. General remarks......Page 265
b. Curvilinear coordinates......Page 270
d. Differentiation formulae for the inverse functions......Page 276
e. Symbolic product of mappings......Page 281
f. General theorem on the inversion of transformations and of systems of implicit functions. Decomposition into primitive mappings......Page 285
g. Alternate construction of the inverse mapping by the method of successive approximations......Page 290
h. Dependent functions......Page 297
i. Concluding remarks......Page 299
a. Elements of the theory of surfaces......Page 302
b. Conformal transformation in general......Page 312
a. General remarks......Page 314
b. Envelopes of one-parameter families of curves......Page 316
c. Examples......Page 320
d. Endevelopes of families of surfaces......Page 327
a. Definition of alternating differential forms......Page 331
b. Sums and products of differential forms......Page 334
c. Exterior derivatives of differential forms......Page 336
d. Exterior differential forms in arbitrary coordinates......Page 340
a. Necessary conditions......Page 349
b. Examples......Page 351
c. Maxima and minima with subsidiary conditions......Page 354
d. Proof of the method of undetermined multipliers in the simplest case......Page 358
e. Generalization of the method of undetermined multipliers......Page 361
f. Examples......Page 364
A.1 Sufficient Conditions for Extreme Values......Page 369
A.2 Numbers of Critical Points Related to Indices of a Vector Field......Page 376
A.3 Singular Points of Plane Curves......Page 384
A.4 Singular Points of Surfaces......Page 386
A.5 Connection Between Euler's and Lagrange's Representation of the motion of a Fluid......Page 387
A.6 Tangential Representation of a Closed Curve and the Isoperimetric Inequality......Page 389
a. Definition of the Jordan measure of area......Page 391
b. A set that does not have an area......Page 394
c. Rules for operations with areas......Page 396
a. The double integral as a volume......Page 398
b. The general analytic concept of the integral......Page 400
c. Examples......Page 403
d. Notation. Extensions. Fundamental rules......Page 405
e. Integral estimates and the mean value theorem......Page 407
4.3 Integrals over Regions in three and more Dimensions......Page 409
4.4 Space Dmerentiation. Mass and Density......Page 410
a. Integrals over a rectangle......Page 412
b. Change of order of integration. Differentiation under the integral sign......Page 414
c. Reduction of double integrals to single integrals for more general regions......Page 416
d. Extension of the results to regions in several dimensions......Page 421
a. Transformation of integrals in the plane......Page 422
b. Regions of more than two dimensions......Page 427
4.7 Improper Multiple Integrals......Page 430
a. Improper integrals of functions over bounded sets......Page 431
b. Proof of the general convergence theorem for improper integrals......Page 435
c. Integrals over unbounded regions......Page 438
a. Elementary calculation of volumes......Page 441
b. General remarks on the calculation of volumes. Solids of revolution. Volumes in spherical coordinates......Page 443
c. Area of a curved surface......Page 445
a. Moments and center of mass......Page 455
b. Moments of inertia......Page 457
c. The compound pendulum......Page 460
d. Potential of attracting masses......Page 462
a. Resolution of multiple integrals......Page 469
b. Application to areas swept out by moving curves and volumes swept out by moving surfaces. Guldin's formula. The polar planimeter......Page 472
a. Surface areas and surface integrals in more than three dimensions......Page 477
b. Area and volume of the n-dimensional sphere......Page 479
c. Generalizations. Parametric Representations......Page 483
a. Uniform convergence. Continuous dependence on the parameter......Page 486
b. Integration and differentiation of improper integrals with respect to a parameter......Page 490
c. Examples......Page 493
d. Evaluation of Fresnel's integrals......Page 497
a. Introduction......Page 500
b. Examples......Page 503
c. Proof of Fourier's integral theorem......Page 505
d. Rate of convergence in Fourier's integral theorem......Page 509
e. Parseval's identity for Fourier transforms......Page 512
f. The Fourier transformation for functions of several variables......Page 514
a. Definition and functional equation......Page 521
b. Convex functions. Proof of Bohr and Mollerup's theorem......Page 523
c. The infinite products for the gamma function......Page 527
d. The nextensio theorem......Page 531
e. The beta function......Page 532
f. Differentiation and integration of fractional order. Abel's integral equation......Page 535
a. Subdivisions of the plane and the corresponding inner and outer areas......Page 539
b. Jordan-measurable sets and their areas......Page 541
c. Basic properties of areas......Page 543
a. Definition of the integral of a function {(x, y)......Page 548
b. Integrability of continuous functions and integrals over sets......Page 550
c. Basic rules for multiple integrals......Page 552
d. Reduction of multiple integrals to repeated single integrals......Page 555
a. Mappings of sets......Page 558
b. Trans formation of multiple integrals......Page 563
A.4 Note on the Definition of the Area of a Curved Surface......Page 564
5.1 Connection Between Line Integrals and Double Integrals in the Plane (The Integral Theorems of Gauss, Stokes, and Green)......Page 567
5.2 Vector Form of the Divergence Theorem. Stokes's Theorem......Page 575
5.3 Formula for Integration by Parts in Two Dimensions. Green's Theorem......Page 580
a. The case of 1-1 mappings......Page 582
b. Transformation of integrals and degree of mapping......Page 585
5.5 Area Differentiation. Transformation of Au to Polar Coordinates......Page 589
5.6 Interpretation of the Formulae of Gauss and Stokes by TwoDimensional Flows......Page 593
a. Orientation of two-dimensional surfaces in three-space......Page 599
b. Orientation of curves on oriented surfaces......Page 611
a. Double integrals over oriented plane regions......Page 613
b. Surface integrals of second-order differential forms......Page 616
c. Relation between integrals of differential forms over oriented surfaces to integrals of scalars over unoriented surfaces......Page 618
a. Gauss's theorem......Page 621
b. Application of Gauss's theorem to fluid flow......Page 626
c. Gauss's theorem applied to space forces and surface forces......Page 629
d. Integration by parts and Green's theorem in three dimensions......Page 631
e. Application of Green's theorem to the transformation of U to spherical coordinates......Page 632
a. Statement and proof of the theorem......Page 635
b. Interpretation of Stokes's theorem......Page 639
5.11 Integral Identities in Higher Dimension......Page 646
a. Elementary surfaces......Page 648
b. Integral of a function over an elementary surface......Page 651
c. Oriented elementary surfaces......Page 653
d. Simple surfaces......Page 655
e. Partitions of unity and integrals over simple surfaces......Page 658
a. Statement of the theorem and its invariance,......Page 661
b. Proof of the theorem,......Page 663
A.3 Stokes's Theorem......Page 666
a. Elementary surfaces......Page 669
b. Integral of a differential form over an oriented elementary surface......Page 671
c. Simple m-dimensional surfaces......Page 672
A.5 Integrals over Simple Surfaces, Gauss's Divergence Theorem, and the General Stokes Formula in Higher Dimensions......Page 675
a. The equations of motion......Page 678
b. The principle of conservation of energy......Page 680
c. Equilibrium. Stability......Page 683
d. Small oscillations about a position of equilibrium......Page 685
e. Planetary motion......Page 689
f. Boundary value problems. The loaded cable and the loaded beam......Page 696
a. Separation of variables......Page 702
b. The linear first..order equation......Page 704
a. Principle of superposition. General solutions......Page 707
b. Homogeneous differential equations of the second second order......Page 712
c. The nonhomogeneous differential equations. Method of variation of parameters......Page 715
a. Geometrical interpretation......Page 721
b. The differential equation of a family of curves. Singular solutions. Orthogonal trajectories......Page 724
c. Theorem of the existence and uniqueness of the solution......Page 726
6.5 Systems of Differential Equations and Differential Equations of Higher Orde......Page 733
6.6 Integration by the Method of Undermined Coefficients......Page 735
a. Potentials of mass distributions......Page 737
b. The differential equation of the potential......Page 742
c. Uniform double layers......Page 743
d. The mean value theorem......Page 746
e. Boundary value problem for the circle. Poisson's integral......Page 748
a. The wave equation in one dimension......Page 751
b. The wave equation in three-dimensional space......Page 752
c. Maxwell's equations in free space......Page 755
7.1 Functions and Their Extrema......Page 761
a. Vanisbing of the first variation......Page 765
b. Deduction of Euler's differential equation......Page 767
c. Proofs of the fundamental lemmas......Page 771
d. Solution of Euler's differential equation in special cases. Examples......Page 772
e. Identical vanishing of Euler's expression......Page 776
a. Integrals with more than one argument function......Page 777
b. Examples......Page 779
c. Hamilton's principle. Lagrange's equations......Page 781
d. Integrals involving higher derivatives......Page 783
e. Several independent variables......Page 784
a. Ordinary subsidiary conditions......Page 786
b. Other types of subsidiary conditions......Page 789
a. Limits and infinite series with complex terms......Page 793
b. Power series......Page 796
c. Differentiation and integration of power series......Page 797
d. Examples of power series......Page 800
a. The postulate of differentiability......Page 802
b. The simplest operations of the differential calculus......Page 806
c. Conformal transformation. Inverse functions......Page 809
a. Definition of the integral......Page 811
b. Cauchy's theorem......Page 813
c. Applications. The logarithm, the exponential function, and the general power function......Page 816
a. Cauchy's formula......Page 821
b. Expansion of analytic functions in power series......Page 823
c. The theory of functions and potential theory......Page 826
e. Zeros, poles, and residues of an analytic function......Page 827
a. Proof of the formula (8.22)......Page 831
b. Proof of the formula (8.23)......Page 832
c. Application of the theorem of residues to the integration of rational functions......Page 833
d. The theorem of residues and linear differential equations with constant coefficients......Page 836
8.6 Many-Valued Functions and Analytic Extension......Page 838
Exercises 1.1 (p.10)......Page 845
Exercises 1.2 (p.16)......Page 846
Exercises 1.3 (p.24)......Page 847
Exercises 1.4a (p.30)......Page 849
Exercises 1.4c (p.36)......Page 851
Problems 1.5a (p.43)......Page 852
Exercises 1.5c (p.48)......Page 853
Exercises 1.5e (p.53)......Page 854
Exercises 1.6a (p.57)......Page 855
Problems 1.6a (p.58)......Page 856
Problems 1.6c (p.64)......Page 857
Exercises 1.7b (p.68)......Page 858
Exercises 1.7c (p.70)......Page 859
Problems 1.7c (p.70)......Page 860
Exercises 1.9b (p.92)......Page 861
Exercises 2.1 (p.141)......Page 862
Exercises 2.2 (p.158)......Page 864
Exercises 2.3 (p.177)......Page 866
Exercises 2.4 (p.202)......Page 869
Exercises 2.5 (p.215)......Page 874
Exercises 3.1a (p.219)......Page 879
Exercises 3.1c (p.225)......Page 880
Exercises 3.1e (p.230)......Page 881
Exercises 3.2a (p.235)......Page 882
Exercises 3.2b (p.237)......Page 883
Exercises 3.3a (p.248)......Page 884
Exercises 3.3d (p.255)......Page 885
Exercises 3.3e (p.260)......Page 888
Exercises 3.3i (p.277)......Page 889
Exercises 3.4a (p.286)......Page 890
Exercises 3.4b (p.289)......Page 892
Exercises 3.5c (p.302)......Page 894
Exercises 3.5d (p.306)......Page 896
Exercises 3.6b (p.312)......Page 897
Exercises 3.6c (p.316)......Page 898
Exercises 3.7c (p.334)......Page 899
Exercises 3.7e (p.340)......Page 900
Exercises 3.7f (p.344)......Page 901
Exercises A.1 (p.350)......Page 903
Exercises A.2 (p.359)......Page 905
Exercises A.5 (p.364)......Page 906
Exercises 4.1 (p.374)......Page 907
Exercises 4.6 (p.405)......Page 910
Exercises 4.8 (p.430)......Page 912
Exercises 4.9 (p.442)......Page 913
Exercises 4.11 (p.462)......Page 917
Exercises 4.12 (p.474)......Page 918
Exercises 4.13 (p.497)......Page 919
Exercises 4.14 (p.513)......Page 920
Exercises 5.2 (p.555)......Page 924
Exercises 5.9e (p.610)......Page 925
Exercises 5.10b (p.617)......Page 927
Exercises 6.1e (p.671)......Page 930
Exercises 6.2 (p.6825)......Page 934
Exercises 6.3b (p.690)......Page 935
Exercises 6.3c (p.695)......Page 938
Exercises 6.4 (p.706)......Page 939
Exercises 6.5 (p.710)......Page 940
Exercises 6.7 (p.726)......Page 942
Exercises 6.8 (p.734)......Page 943
Exercises 7.2d (p.751)......Page 947
Exercises 7.3b (p.757)......Page 948
Exercises 7.4a (p.765)......Page 949
Exercises 7.4b (p.767)......Page 950
Exercises 8.1 (p.777)......Page 951
Exercises 8.2 (p.786)......Page 952
Exercises 8.3 (p.796)......Page 953
Exercises 8.4 (p.805)......Page 956
Exercises 8.5 (p.814)......Page 959
Miscellaneous Exercises 8 (p.818)......Page 960
List of Biographical Dates......Page 965
Index......Page 967