费曼强烈推荐的经典教材_Advance_Calculus_by_woods

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PREFACE The Fourier series are introduced later as tools for solving certain partial differential equations, but no attempt has been made to develop their theory The subjects treated in the book may be. most easily seen by examining the table of contents. Experience has shown that the book may be covered in a years course FREDERICK S WOODS NoTE FOR THE 1982 PRINTING. In this impression of the book certain improvements have been made. In particular, Osgood's theorem has been inserted in Chapter I, the discussion of uniform convergence in Chapter II has been improved, and the treatment of the plane in Chapter V has been changed PREFACE TO THE NEW EDITION In this edition additional exercises have been inserted at the end of most chapters. Also, in Chapter VI, certain proofs have been made more rigorous namely, that for the existence of the definite integral and that for the possibility of differentiating under the integral sign a definite integral with upper limit infinity. All the typographical errors that have been discovered have been corrected FREDERICK S WOODS CONTENTS CHAPTER I. PRELIMINARY SECTION PAGE 1. Functiong 2. Continuity 3. The derivative 4. Composite functions 5. Rolle' s theorem 125778 6. Theorem of the mean d. Taylors series with a remainder 10 8. The form 15 0 9. The form 16 10. Other indeterminate forms 18 11. Infinitesimals 19 12. Fundamental theorems on infinitesimals 22 13. Some geometric theorems involving infinitesimals 23 14. The first differential 28 15. Higher differentials 29 16. Change of variable 92 CHAPTER II. POWER SERIES 17. Definitions 88 18. Comparison test for convergence 。,,,,40 19. The ratio test for convergence 41 20 Region of convergence 42 21. Uniform convergenee 45 22. Function defined by a power series 45 23. Integral and derivative of a power series 46 24. Taylor,'s series 48 25 Operations with two power series 51 26. The exponential and trigonometric functions .......... 53 27. Hyperbolic functions 55 28. Dominant functiong 57 29. Conditionally convergent series CHAPTER III. PARTIAL DIFFERENTIATION 80. Functions of two or more variables 65 31. Partial derivatives 66 32. Order of differentiation 68 88 Diferentiation of composite functiong 69 CONTENTS SECTION PAGE 34. Euler@g theorem on homogeneous functions 73 35. Directional derivative 74 6。 The frst differential 78 37. Higher differentials 84 38. Taylor's serie 85 CHAPTER IV. IMPLICIT FUNCTIONS 89. One equation, two variables 91 40, One equation, more than two variables 93 41. Two equations, four variables 95 42. Three equations, six variables 97 43. The general case 98 44。 Jacobians, 99 CHAPTER V. APPLICATIONS TO GEOMETRY 45。 lement of arc., 106 46. Straight line 108 47 Surfaces 109 48. planes 110 49. Behavior of a surface near a point ,112 50. Maxima and minima 116 51. Curves 118 52. Curvature and torsion 121 53, Curvilinear coordinates 124 CHAPTER VI. THE DEFINITE INTEGRAL 54. Derinition 134 56. Existence proof 135 56. Properties of definite integrals 137 57. Evaluation of a definite integral 138 58. Simpson's rule 39. Change of variableg 139 ,。140 60, Differentiation of a definite integral 141 61. Integration under the integral sign 145 62. Infinite limit 146 68. Differentiation and integration of an integral with an infinite limit. 148 64. Infinite integrand 161 65. Certain definite integrals 153 66. Multiple integrals 156 CHAPTER VII. THE GAMMA AND BETA FUNCTIONS 67. The Gamma function 164 68. The Beta function 166 69. Dirichlet's integrals 167 70. Special relations 169 CONTENTS CHAPTER XVI. ELLIPTIC INTEGRALS BGION PAGE 161. Introduction 366 152. The functions sn u, cn u, dn u 367 150. Application to the pendulum 369 164. Formulas of differentiation and series expansion 371 155. Addition formulas 372 156. The periods 873 157.: cases 376 158 Elliptic integrals in the complex plane 376 169. Elliptic integrals of the second kind and of the third kind 879 160. The function p(u) 881 161. Applications 382 ANSWERS 887 INDEX 395 ADVANCED CALCULUS CHAPTER I PRELIMINARY 1. Functions. A quantity y is said to be a function of a nuan tity a if the value of y i8 determined when the value of a is given. Elementary examples are the familiar algebraic, trigonometric logarithmic, and exponential functions by means of which y is explicitly given in terms of a. Such explicit formulation, how ever, is not necessary to the idea of a function For example, y may be the number of cents of postage on a letter and the number of ounces in its weight or y may be defined as the largest prime number which is smaller than any number a, or y may be defined as equal to 0 if a is a rational number and equal to l if a is an irrationel number. It should be - noticed, moreover, that even when an explicit formulation in elementary functions is possible, y need not be defined by the same formula for all values of n. For example, consider a spherical shell of inner radius a and outer radius b composed of matter of density p. Let a be the dis- ance of a point from the center of the shell and y the gravita- tional potential due to the shell. Then y is a function of a with the following formulation y=2 p(b-a when 2 a v=2丌p(b2 Tp when芝 4丌 8 (b3-a)when a >b o we may at pleasure build up an arbitrary function of For example, let y=f(a), where f(c)=2x2 when 0 <x <1, f(x)=景 when 0=1, f(x)=景x+1 when x>1 2 PRELⅠ MINARY We shall say that values of a which lie between a and b deter- mine an interval (a, 6). The interval may or may not include the values a and b, according to the con- text. In general, however the inter- vals (a, b) will mean the values of a defined by the statement a≡≡b The student is supposed to be fa- miliar with the representation of a function by a graph. Such a repre- sentation is usually possible for theO C functions we shall handle in this book FIG. 1 although it is impossible for the func- tion mentioned in the third example of this section. The in- terval(a, b)appears in the graph as the portion of the axis of between a=a and =b, and it will be Y convenient to speak of a point of the interval, meaning a value of u in the interval. Then r=a and x=b are the end-points of the interval. As men- tioned above, the interval may or may not have end-points. The graph of the potential function in (1 is the curve of Fig 1. The graph has O x no breaks and the function is continuous FIG 2 (9 2), but the character of the curve and of the function is different in the three intervals considered The graph of the function in(2) is the curve of Fig. 2. This graph has a break at the point for which x= 1. 2. Continuity. A function f(a)is continuous when a =a for which fla is defined if Iim[f(+b)一f()]=0, 0 or, otherwise expressed, if Limf(a+h)=f(a) h→0 where in either formula the limit is independent of the manner in which h approaches 0 Since h is an increment added to a, and f(a+h-f(a is the corresponding increment of f(a), we may express this definition as follows A function of r is continuous for a given value of a if the increment of the function approaches zero as the increment of a approaches zero. CONTINUITY 3 A more cumbersome definition, but one which brings out the full meaning of equation(1), is as follows f(a)is continuous for a=a when if e is any assigned positive quantity, no matter how small, it is possible to determine another positive antity 8 so-that the difference in absolute valae between f(a+ h)and f(a) shall be less than e for all values of h numerically less than 8, that is I f(a+h)-f(a)I<E when h <8 (3) Graphically, e having been given, there can be found an interval (a+h,a-h)in which I f(ec)-f(a)I<e at all points of the interval Consider the function defined by the equations 10 when a≠0, 1+ ∫(0)=0, the graph of which is shown in Fig 3. Here f(0+h)f(o)when h ap proaches zero through positive values FIG. 3 andf(0+b)→10≠f(0) when h ap- proaches zero through negative values. Hence the function is not continuous when r=0. There is no interval (h, h in which I f(a)-f(0) <E. Furthermore, while the definition of f(o) in(4)is arbi- trary, it is not possible to define f(o) so that the function is continuous It is to be noticed that f()is not continuous for a s a if f(a) is infi- nite. ' This expression means that x f(a+ h)can be made numerically larger than any assigned positive quantity by taking h sufficiently small; or, more precisely, if M is a positive number no matter how large then a FIG. 4 number s can be determined so that I f(a+h)I>M for h <8. The definition of cotinuity cannot then be satisfied for =a. 1 For example, the functions -(Fig 4)and -a(Fig. 5)are each discontinuous for r=0. as is shown by the break in each of the curves representing the functions. PRELIMINARY The following theorems are of fundamental importance in handling continuous functions 1. If f(a) is continuous at all points of an interval (a, b, it is possible to find a positive number s such that in all subin tervals of(a, b)less than s the absolute value of the difference betwveen any two values of f(ec) is less than E when E is a positive quantity given in advance. We shall not give a formal proof. It s not difficult to see that if these theo rems were not true, definition 3)for continuity must fail for at least one point of (a, b). Because of the property X stated in the theorem, f(a)is said to be uniformly continuous in(a, b) FIG. 5 II. If f(e)is continuous for all values of a between a and b inclu Y sive, if f(a)=A and f(b)=B, and if n is any value betueen A and B, then f(a)= N for at least one value of s between a andb. B N M FIG. 6 FIG. 7 IIL. If f(e)is continuous for all values of a between a and b inclu sive, then f(e)has a largest value M for at least one value of a between a Y and b and a smallest value m for at least some other value of a between a and b These theorems seem to be inherent in the very nature of continuity and are graphically evident from Figs. 6. 7. and 8. As a matter of fact. how- X ever, they are not self-evident and FIG8

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