This book is a basic text in advanced calculus, providing a clear and well motivated,
yet precise and rigorous, treatment of the essential tools of
mathematical analysis at a level immediately following that of a first course
in calculus. It is designed to satisfy many needs; it fills gaps that almost
always, and properly, occur in elementary calculus courses; it contains all
of the material in the standard classical advanced calculus course; and it
provides a solid foundation in the "deltas and epsilons" of a modern rigorous
advanced calculus. It is well suited for courses of considerable diversity,
ranging from "foundations of calculus" to "critical reasoning in mathematical
analysis." There is even ample material for a course having a standard
advanced course as prerequisite.
Throughout the book attention is paid to the average or less-than-average
student as well as to the superior student. This is done at every stage of
progress by making maximally available whatever concepts and discussion
are both relevant and understandable. To illustrate: limit and continuity
theorems whose proofs are difficult are discussed and worked with before
they are proved, implicit functions are treated before their existence is
established, and standard power series techniques are developed before the
topic of uniform convergence is studied. Whenever feasible, if both an
elementary and a sophisticated proof of a theorem are possible, the elementary
proof is given in the text, with the sophisticated proof possibly called for in
an exercise, with hints. Generally speaking, the more subtle and advanced
portions of the book are marked with stars ( *), prerequisite for which is
preceding starred material. This contributes to an unusual flexibility of the
book as a text.
The author believes that most students can best appreciate the more difficult
and advanced aspects of any field of study if they have thoroughly mastered
the relatively easy and introductory parts first. In keeping with this philosophy,
the book is arranged so that progress moves from the simple to the
complex and from the particular to the general. Emphasis is on the concrete,
with abstract concepts introduced only as they are relevant, although the
general spirit is modern. The Riemann integral, for example, is studied first
with emphasis on relatively direct consequences of basic definitions, and
then with more difficult results obtained with the aid of step functions.
Later some of these ideas are extended to multiple integrals and to the
Riemann-Stieltjes integral. Improper integrals are treated at two levels of
sophistication; in Chapter 4 the principal ideas are dominance and the
"big 0" and "little o" concepts, while in Chapter 14 uniform convergence
becomes central, with applications to such topics as evaluations and the
gamma and beta functions.
Vectors are presented in such a way that a teacher using this book may
almost completely avoid the vector parts of advanced calculus if he wishes
to emphasize the "real variables" content. This is done by restricting the
use of vectors in the main part of the book to the scalar, or dot, product,
with applications to such topics as solid analytic geometry, partial differentiation,
and Fourier series. The vector, or cross, product and the differential
and integral calculus of vectors are fully developed and exploited in the last
three chapters on vector analysis, line and surface integrals, and differential
geometry. The now-standard Gibbs notation is used. Vectors are designated
by means of arrows, rather than bold-face type, to conform with
handwriting custom.
Special attention should be called to the abundant sets of problems-there
are over 2440 exercises! These include routine drills for practice, intermediate
exercises that extend the material of the text while retaining its
character, and advanced exercises that go beyond the standard textual subject
matter. Whenever guidance seems desirable, generous hints are included.
In this manner the student is led to such items of interest as limits superior
and inferior, for both sequences and real-valued functions in general, the
construction of a continuous nondifferentiable function, the elementary
theory of analytic functions of a complex variable, and exterior differential
forms. Analytic treatment of the logarithmic, exponential, and trigonometric
functions is presented in the exercises, where sufficient hints are given to
make these topics available to all. Answers to all problems are given in the
back of the book. Illustrative examples abound throughout.
Standard Aristotelian logic is assumed; for example, frequent use is made
of the indirect method of proof. An implication of the form p implies q is
taken to mean that it is impossible for p to be true and q to be false simultaneously;
in other words, that the conjunction of the two statements p
and not q leads to a contradiction. Any statement of equality means simply
that the two objects that are on opposite sides of the equal sign are the same
thing. Thus such statements as "equals may be added to equals," and "two
things equal to the same thing are equal to each other," are true by definition.
A few words regarding notation should be given. The equal sign == is used
for equations, both conditional and identical, and the triple bar - is reserved
for definitions. For simplicity, if the meaning is clear from the context,
neither symbol is restricted to the indicative mood as in "(a + b )2 ==
a2 + 2ab + b2," or "where f(x) - x2 + 5." Examples of subjunctive uses
are "let x == n," and "let e - 1," which would be read "let x be equal ton,"
and "let e be defined to be 1," respectively. A similar freedom is granted
the inequality symbols. For instance, the symbol > in the following constructions
"if e > 0, then · · · ," "let e > 0," and "let e > 0 be given,"
could be translated "is greater than," "be greater than," and "greater than,"
respectively. A relaxed attitude is also adopted regarding functional
notation, and the tradition (y == f(x)) established by Dirichlet has been
followed. When there can be no reasonable misinterpretation the notation
f(x) is used both to indicate the value of the function f corresponding to a
particular value x of the independent variable and also to represent the
function f itself (and similarly for f(x, y), f(x, y, z), and the like). This
permissiveness has two merits. In the first place it indicates in a simple way
the number of independent variables and the letters representing them. In
the second place it avoids such elaborate constructions as "the function f
defined by the equation f(x) == sin 2x is periodic," by permitting simply,
"sin 2x is periodic." This practice is in the spirit of such statements as "the
line x + y == 2 · · · ," instead of "the line that is the graph of the equation
x + y == 2 · · ·," and "this is John Smith," instead of "this is a man whose
name is John Smith."
In a few places parentheses are used to indicate alternatives. The principal
instances of such uses are heralded by announcements or footnotes in the
text. Here again it is hoped that the context will prevent any ambiguity.
Such a sentence as "The function j"(x) is integrable from a to b (a < b)"
would mean that ''f(x) is integrable from a to b, where it is assumed that
a < b," whereas a sentence like "A function having a positive (negative)
derivative over an interval is strictly increasing (decreasing) there" is a
compression of two statements into one, the parentheses indicating an
alternative formulation.
Although this text is almost completely self-contained, it is impossible
within the compass of a book of this size to pursue every topic to the extent
that might be desired by every reader. Numerous references to other books
are inserted to aid the intellectually ambitious and curious. Since many of
these references are to the author's Real Variables (abbreviated here to RV),
of this same Appleton-Century Mathematics Series, and since the present
Advanced Calculus (AC for short) and RV have a very substantial body of
common material, the reader or potential user of either book is entitled to
at least a short explanation of the differences in their objectives. In brief,
A C is designed principally for fairly standard advanced calculus courses, of
either the "vector analysis" or the "rigorous" type, while RV is designed
principally for courses in introductory real variables at either the advanced
calculus or the post-advanced calculus level. Topics that are in both AC and
RV include all those of the basic "rigorous advanced calculus." Topics that
viii PREFACE
are in AC but not in RV include solid analytic geometry, vector analysis,
complex variables, extensive treatment of line and surface integrals, and
differential geometry. Topics that are in RVbut not in ACinclude a thorough
treatment of certain properties of the real numbers, dominated convergence
and measure zero as related to the Riemann integral, bounded variation as
related to the Riemann-Stieltjes integral and to arc length, space-filling arcs,
independence of parametrization for simple arc length, the Moore-Osgood
uniform convergence theorem, metric and topological spaces, a rigorous
proof of the transformation theorem for multiple integrals, certain theorems
on improper integrals, the Gibbs phenomenon, closed and complete orthonormal
systems of functions, and the Gram-Schmidt process.
One note of caution is in order. Because of the rich abundance of material
available, complete coverage in one year is difficult. Most of the unstarred
sections can be completed in a year's sequence, but many teachers will wish
to sacrifice some of the later unstarred portions in order to include some of
the earlier starred items. Anybody using the book as a text should be advised
to give some advance thought to the main emphasis he wishes to give his
course and to the selection of material suitable to that emphasis.
Author(s): John M. H. Olmsted
Edition: 1
Publisher: Appleton-Century-Crofts
Year: 1961
Language: English
Pages: 706
PREFACE
CONTENTS
Chapter I THE REAL NUMBER SYSTEM
101. Introduction
102. Axioms of a field
103. Exercises
104. Axioms of an ordered field
105. Exercises
106. Positive integers and mathematical induction
107. Exercises
108. Integers and rational numbers
109. Exercises
110. Geometrical representation and absolute value
111. Exercises
112. Axiom of completeness
113. Consequences of completeness
114. Exercises
Chapter 2 FUNCTIONS, SEQUENCES, LIMITS, CONTINUITY
201. Functions and sequences
202. Limit of a sequence
203. Exercises
204. Limit theorems for sequences
205. Exercises
206. Limits of functions
207. Limit theorems for functions
208. Exercises
209. Continuity
210. Types of discontinuity
211. Continuity theorems
212. Exercises
213. More theorems on continuous functions
214. Existence of √2 and other roots
215. Monotonic functions and their inverses
216. Exercises
*217. A fundamental theorem on bounded sequences
*218. Proofs of some theorems on continuous functions
*219. The Cauchy criterion for convergence of a sequence
*220. Exercises
*221. Sequential criteria for continuity and existence of limits
*222. The Cauchy criterion for functions
*223. Exercises
*224. Uniform continuity
*225. Exercises
Chapter 3 DIFFERENTIATION
301. Introduction
302. The derivative
303. One-sided derivatives
304. Exercises
305. Rolle's theorem and the Law of the Mean
306. Consequences of the Law of the Mean
307. The Extended Law of the Mean
308. Exercises
309. Maxima and minima
310. Exercises
311. Differentials
312. Approximations by differentials
313. Exercises
314. L'Hospital's Rule. Introduction
315. The indeterminate form 0/0
316. The indeterminate form ∞/∞
317. Other indeterminate forms
318. Exercises
319. Curve tracing
320. Exercises
*321. Without loss of generality
*322. Exercises
Chapter 4 INTEGRATION
401. The definite integral
402. Exercises
*403. More integration theorems
*404. Exercises
405. The Fundamental Theorem of Integral Calculus
406. Integration by substitution
407. Exercises
408. Sectional continuity and smoothness
409. Exercises
410. Reduction formulas
411. Exercises
412. Improper integrals, introduction
413. Improper integrals, finite interval
414. Improper integrals, infinite interval
415. Comparison tests. Dominance
416. Exercises
*417. The Riemann-Stieltjes integral
*418. Exercises
Chapter 5 SOME ELEMENTARY FUNCTIONS
*501. The exponential and logarithmic functions
*502. Exercises
*503. The trigonometric functions
*504. Exercises
505. Some integration formulas
506. Exercises
507. Hyperbolic functions
508. Inverse hyperbolic functions
509. Exercises
*510. Classification of numbers and functions
*511. The elementary functions
*512. Exercises
Chapter 6 FUNCTIONS OF SEVERAL VARIABLES
601. Introduction
602. Neighborhoods in the Euclidean plane
603. Point sets in the Euclidean plane
604. Sets in higher-dimensional Euclidean spaces
605. Exercises
606. Functions and limits
607. Iterated limits
608. Continuity
609. Limit and continuity theorems
610. More theorems on continuous functions
611. Exercises
612. More general functions. Mappings
*613. Sequences of points
*614. Point sets and sequences
*615. Compactness and continuity
*616. Proofs of two theorems
*617. Uniform continuity
618. Exercises
Chapter 7 SOLID ANALYTIC GEOMETRY AND VECTORS
701. Introduction
702. Vectors and scalars
703. Addition and subtraction of vectors. Magnitude
704. Linear combinations of vectors
705. Exercises
706. Direction angles and cosines
707. The scalar or inner or dot product
708. Vectors orthogonal to two vectors
709. Exercises
710. Planes
711. Lines
712. Exercises
713. Surfaces. Sections, traces, intercepts
714. Spheres
715. Cylinders
716. Surfaces of revolution
717. Exercises
718. The standard quadric surfaces
719. Exercises
Chapter 8 ARCS AND CURVES
801. Duhamel's principle for integrals
*802. A proof with continuity hypotheses
803. Arcs and curves
804. Arc length
805. Integral form for arc length
*806. Remark concerning the trigonometric functions
807. Exercises
808. Cylindrical and spherical coordinates
809. Arc length in rectangular, cylindrical, and spherical coordinates
810. Exercises
811. Curvature and radius of curvature in two dimensions
812. Circle of curvature
*813. Evolutes and involutes
814. Exercises
Chapter 9 PARTIAL DIFFERENTIATION
901. Partial derivatives
902. Partial derivatives of higher order
*903. Equality of mixed partial derivatives
904. Exercises
905. The fundamental increment formula
906. Differentials
907. Change of variables. The chain rule
*908. Homogeneous functions. Euler's theorem
909. Exercises
*910. Directional derivatives. Tangents and normals
*911. Exercises
912. The Law of the Mean
913. Approximations by differentials
914. Maxima and minima
915. Exercises
916. Differentiation of an implicit function
917. Some notational pitfalls
918. Exercises
919. Envelope of a family of plane curves
920. Exercises
921. Several functions defined implicitly. Jacobians
922. Coordinate transformations. Inverse transformations
923. Functional dependence
924. Exercises
925. Extrema with one constraint. Two variables
926. Extrema with one constraint. More than two variables
927. Extrema with more than one constraint
928. Lagrange multipliers
929. Exercises
*930. Differentiation under the integral sign. Leibnitz's rule
*931. Exercises
*932. The Implicit Function Theorem
*933. Existence theorem for inverse transformations
*934. Sufficiency conditions for functional dependence
*935. Exercises
Chapter 10 MULTIPLE INTEGRALS
1001. Introduction
1002. Double integrals
1003. Area
1004. Second formulation of the double integral
*1005. Inner and outer area. Criterion for area
*1006. Theorems on double integrals
*1007. Proof of the second formulation
1008. Iterated integrals, two variables
*1009. Proof of the Fundamental Theorem
1010. Exercises
1011. Triple integrals. Volume
1012. Exercises
1013. Double integrals in polar coordinates
1014. Volumes with double integrals in polar coordinates
1015. Exercises
1016. Mass of a plane region of variable density
1017. Moments and centroid of a plane region
1018. Exercises
1019. Triple integrals, cylindrical coordinates
1020. Triple integrals, spherical coordinates
1021. Mass, moments, and centroid of a space region
1022. Exercises
1023. Mass, moments, and centroid of an arc
1024. Attraction
1025. Exercises
1026. Jacobians and transformations of multiple integrals
1027. General discussion
1028. Exercises
Chapter 11 INFINITE SERIES OF CONSTANTS
1101. Basic definitions
1102. Three elementary theorems
1103. A necessary condition for convergence
1104. The geometric series
1105. Positive series
1106. The integral test
1107. Exercises
1108. Comparison tests. Dominance
1109. The ratio test
1110. The root test
1111. Exercises
*1112. More refined tests
*1113. Exercises
1114. Series of arbitrary terms
1115. Alternating series
1116. Absolute and conditional convergence
1117. Exercises
1118. Groupings and rearrangements
1119. Addition, subtraction, and multiplication of series
*1120. Some aids to computation
1121. Exercises
Chapter 12 POWER SERIES
1201. Interval of convergence
1202. Exercises
1203. Taylor series
1204. Taylor's formula with a remainder
1205. Expansions of functions
1206. Exercises
1207. Some Maclaurin series
1208. Elementary operations with power series
1209. Substitution of power series
1210. Integration and differentiation of power series
1211. Exercises
1212. Indeterminate expressions
1213. Computations
1214. Exercises
1215. Taylor series, several variables
1216. Exercises
*Chapter 13 UNIFORM CONVERGENCE AND LIMITS
*1301. Uniform convergence of sequences
*1302. Uniform convergence of series
*1303. Dominance and the Weierstrass M-test
*1304. Exercises
*1305. Uniform convergence and continuity
*1306. Uniform convergence and integration
*1307. Uniform convergence and differentiation
*1308. Exercises
*1309. Power series. Abel's theorem
*1310. Proof of Abel's theorem
*1311. Exercises
*1312. Functions defined by power series. Exercises
*1313. Uniform limits of functions
*1314. Three theorems on uniform limits
*1315. Exercises
*Chapter 14 IMPROPER INTEGRALS
*1401. Introduction. Review
*1402. Alternating integrals. Abel's test
*1403. Exercises
*1404. Uniform convergence
*1405. Dominance and the Weierstrass M-test
*1406. The Cauchy criterion and Abel's test for uniform convergence
*1407. Three theorems on uniform convergence
*1408. Evaluation of improper integrals
*1409. Exercises
*1410. The gamma function
*1411. The beta function
*1412. Exercises
*1413. Infinite products
*1414. Wallis's infinite product for π
*1415. Euler's constant
*1416. Stirling's formula
*1417. Weierstrass's infinite product for 1/Γ(α)
*1418. Exercises
*1419. Improper multiple integrals
*1420. Exercises
Chapter 15 COMPLEX VARIABLES
1501. Introduction
1502. Complex numbers
1503. Embedding of the real numbers
1504. The number i
1505. Geometrical representation
1506. Polar form
1507. Conjugates
1508. Roots
1509. Exercises
1510. Limits and continuity
1511. Sequences and series
1512. Exercises
1513. Complex-valued functions of a real variable
1514. Exercises
*1515. The Fundamental Theorem of Algebra
Chapter 16 FOURIER SERIES
1601. Introduction
1602. Linear function spaces
1603. Periodic functions. The space R_2π
1604. Inner product. Orthogonality. Distance
1605. Least squares. Fourier coefficients
1606. Fourier series
1607. Exercises
1608. A convergence theorem. The space S_2π
1609. Bessel's inequality. Parseval's equation
1610. Cosine series. Sine series
1611. Other intervals
1612. Exercises
*1613. Partial sums of Fourier series
*1614. Functions with one-sided limits
*1615. The Riemann-Lebesgue Theorem
*1616. Proof of the convergence theorem
*1617. Fejer's summability theorem
*1618. Uniform summability
*1619. Weierstrass's theorem
*1620. Density of trigonometric polynomials
*1621. Some consequences of density
*1622. Further remarks
*1623. Other orthonormal systems
*1624. Exercises
1625. Applications of Fourier series. The vibrating string
1626. A heat conduction problem
1627. Exercises
Chapter 17 VECTOR ANALYSIS
1701. Introduction
1702. The vector or outer or cross product
1703. The triple scalar product. Orientation in space
1704. The triple vector product
1705. Exercises
1706. Coordinate transformations
1707. Translations
1708. Rotations
1709. Exercises
1710. Scalar and vector fields. Vector functions
1711. Ordinary derivatives of vector functions
1712. The gradient of a scalar field
1713. The divergence and curl of a vector field
1714. Relations among vector operations
1715. Exercises
*1716. Independence of the coordinate system
*1717. Curvilinear coordinates. Orthogonal coordinates
*1718. Vector operations in orthogonal coordinates
*1719. Exercises
Chapter 18 LINE AND SURFACE INTEGRALS
1801. Introduction
1802. Line integrals in the plane
1803. Independence of path and exact differentials
1804. Exercises
1805. Green's Theorem in the plane
1806. Local exactness
1807. Simply- and multiply-connected regions
1808. Equivalences in simply-connected regions
1809. Exercises
*1810. Analytic functions of a complex variable. Exercises
1811. Surface elements
1812. Smooth surfaces
1813. Schwarz's example
1814. Surface area
1815. Exercises
1816. Surface integrals
1817. Orientable smooth surfaces
1818. Surfaces with edges and corners
1819. The divergence theorem
1820. Green's identities
1821. Harmonic functions
1822. Exercises
1823. Orientable sectionally smooth surfaces
1824. Stokes's Theorem
1825. Independence of path. Scalar potential
*1826. Vector potential
1827. Exercises
*1828. Exterior differential forms. Exercises
Chapter 19 DIFFERENTIAL GEOMETRY
1901. Introduction
1902. Curvature. Osculating plane
1903. Applications to kinematics
1904. Torsion. The Frenet formulas
1905. Local behavior
1906. Exercises
1907. Curves on a surface. First fundamental form
1908. Intersections of smooth surfaces
1909. Plane sections. Meusnier's theorem
1910. Normal sections. Mean and total curvature
1911. Second fundamental form
1912. Exercises
ANSWERS TO PROBLEMS
INDEX
