Introduction to Calculus and Analysis

by Courant, Richard; John, Fritz

• ISBN13: 9783540665694
• ISBN10: 3540665692
• Edition: Reprint
• Format: Paperback
• Copyright: 2000-02-01
• Publisher: Springer Verlag

Summary

From the reviews: "These books (Introduction to Calculus and Analysis Vol. I/II) are very well written. The mathematics are rigorous but the many examples that are given and the applications that are treated make the books extremely readable and the arguments easy to understand. These books are ideally suited for an undergraduate calculus course. Each chapter is followed by a number of interesting exercises. More difficult parts are marked with an asterisk. There are many illuminating figures...Of interest to students, mathematicians, scientists and engineers. Even more than that."Newsletter on Computational and Applied Mathematics, 1991"...one of the best textbooks introducing several generations of mathematicians to higher mathematics. ... This excellent book is highly recommended both to instructors and students." Acta Scientiarum Mathematicarum, 1991

Table of Contents

Functions of Several Variables and Their Derivatives
Points and Points Sets in the Plane and in Space	1	(10)
Sequences of points. Convergence	1	(2)
Sets of points in the plane	3	(3)
The boundary of a set. Closed and open sets	6	(3)
Closure as set of limit points	9	(1)
Points and sets of points in space	9	(2)
Functions of Several Independent Variables	11	(6)
Functions and their domains	11	(1)
The simplest types of functions	12	(1)
Geometrical representation of functions	13	(4)
Continuity	17	(9)
Definition	17	(2)
The concept of limit of a function of several variables	19	(3)
The order to which a function vanishes	22	(4)
The Partial Derivatives of a Function	26	(14)
Definition. Geometrical representation	26	(6)
Examples	32	(2)
Continuity and the existence of partial derivatives	34	(2)
Change of the order of differentiation	36	(4)
The Differential of a Function and Its Geometrical Meaning	40	(13)
The concept of differentiability	40	(3)
Directional derivatives	43	(3)
Geometric interpretation of differentiability, The tangent plane	46	(3)
The total differential of a function	49	(3)
Application to the calculus of errors	52	(1)
Functions of Functions (Compound Functions) and the Introduction of New Independent Variables	53	(11)
Compound functions. The chain rule	53	(6)
Examples	59	(1)
Change of independent variables	60	(4)
The Mean Value Theorem and Taylor's Theorem for Functions of Several Variables	64	(7)
Preliminary remarks about approximation by polynomials	64	(2)
The mean value theorem	66	(2)
Taylor's theorem for several independent variables	68	(3)
Integrals of a Function Depending on a Parameter	71	(11)
Examples and definitions	71	(3)
Continuity and differentiability of an integral with respect to the parameter	74	(6)
Interchange of integrations. Smoothing of functions	80	(2)
Differentials and Line Integrals	82	(13)
Linear differential forms	82	(3)
Line integrals of linear differential forms	85	(7)
Dependence of line integrals on endpoints	92	(3)
The Fundamental Theorem on Integrability of Linear Differential Forms	95	(12)
Integration of total differentials	95	(1)
Necessary conditions for line integrals to depend only on the end points	96	(2)
Insufficiency of the integrability conditions	98	(4)
Simply connected sets	102	(2)
The fundamental theorem	104	(3)
APPENDIX
A.1 The Principle of the Point of Accumulation in Several Dimensions and Its Applications	107	(5)
a. The principle of the point of accumulation	107	(1)
b. Cauchy's convergence test. Compactness	108	(1)
c. The Heine-Borel covering theorem	109	(1)
d. An application of the Heine-Borel theorem to closed sets contains in open sets	110	(2)
A.2 Basic Properties of Continuous Functions	112	(1)
A.3 Basic Notions of the Theory of Point Sets	113	(6)
a. Sets and sub-sets	113	(2)
b. Union and intersection of sets	115	(2)
c. Applications to sets of points in the plane	117	(2)
A.4 Homogeneous functions	119	(3)
Vectors, Matrices, Linear Transformations
Operations with Vectors	122	(21)
Definition of vectors	122	(2)
Geometric representation of vectors	124	(3)
Length of vectors. Angles between directions	127	(4)
Scalar products of vectors	131	(2)
Equation of hyperplanes in vector form	133	(3)
Linear dependence of vectors and systems of linear equations	136	(7)
Matrices and Linear Transformations	143	(16)
Change of base. Linear spaces	143	(3)
Matrices	146	(4)
Operations with matrices	150	(3)
Square matrices. The reciprocal of a matrix. Orthogonal matrices	153	(6)
Determinants	159	(21)
Determinants of second and third order	159	(4)
Linear and multilinear forms of vectors	163	(3)
Alternating multilinear forms. Definition of determinants	166	(5)
Principal properties of determinants	171	(4)
Application of determinants to systems of linear equations	175	(5)
Geometrical Interpretation of Determinants	180	(24)
Vector products and volumes of parallelepipeds in three-dimensional space	180	(7)
Expansion of a determinant with respect to a column. Vector products in higher dimensions	187	(3)
Areas of parallelograms and volumes of parallelepipeds in higher dimensions	190	(5)
Orientation of parallelepipeds in n-dimensional space	195	(5)
Orientation of planes and hyperplanes	200	(1)
Change of volume of parallelepipeds in linear transformations	201	(3)
Vector Notions in Analysis	204	(14)
Vector fields	204	(1)
Gradient of a scalar	205	(3)
Divergence and curl of a vector field	208	(3)
Families of vectors. Application to the theory of curves in space and to motion of particles	211	(7)
Developments and Applications of the Differential Calculus
Implicit Functions	218	(12)
General remarks	218	(1)
Geometrical interpretation	219	(2)
The implicit function theorem	221	(4)
Proof of the implicit function theorem	225	(3)
The implicit function theorem for more than two independent variables	228	(2)
Curves and Surfaces in Implicit Form	230	(11)
Plane curves in implicit form	230	(6)
Singular points of curves	236	(2)
Implicit representation of surfaces	238	(3)
Systems of Functions, Transformations, and Mappings	241	(37)
General remarks	241	(5)
Curvilinear coordinates	246	(3)
Extension to more than two independent variables	249	(3)
Differentiation formulae for the inverse functions	252	(5)
Symbolic product of mappings	257	(4)
General theorem on the inversion of transformations and of systems of implicit functions. Decomposition into primitive mappings	261	(5)
Alternate construction of the inverse mapping by the method of successive approximations	266	(2)
Dependent functions	268	(7)
Concluding remarks	275	(3)
Applications	278	(12)
Elements of the theory of surfaces	278	(11)
Conformal transformation in general	289	(1)
Families of Curves, Families of Surfaces, and Their Envelopes	290	(17)
General remarks	290	(2)
Envelopes of one-parameter families of curves	292	(4)
Examples	296	(7)
Endevelopes of families of surfaces	303	(4)
Alternating Differential Forms	307	(18)
Definition of alternating differential forms	307	(3)
Sums and products of differential forms	310	(2)
Exterior derivatives of differential forms	312	(4)
Exterior differential forms in arbitrary coordinates	316	(9)
Maxima and Minima	325	(20)
Necessary conditions	325	(2)
Examples	327	(3)
Maxima and minima with subsidiary conditions	330	(4)
Proof of the method of undetermined multipliers in the simplest case	334	(3)
Generalization of the method of undetermined multipliers	337	(3)
Examples	340	(5)
APPENDIX
A.1 Sufficient Conditions for Extreme Values	345	(7)
A.2 Numbers of Critical Points Related to Indices of a Vector Field	352	(8)
A.3 Singular Points of Plane Curves	360	(2)
A.4 Singular Points of Surfaces	362	(1)
A.5 Connection Between Euler's and Lagrange's Representation of the motion of a Fluid	363	(2)
A.6 Tangenital Representation of a Closed Curve and the Isoperimetric Inequality	365	(2)
Multiple Integrals
Areas in the Plane	367	(7)
Definition of the Jordan measure of area	367	(3)
A set that does not have an area	370	(2)
Rules for operations with areas	372	(2)
Double Integrals	374	(11)
The double integral as a volume	374	(2)
The general analytic concept of the integral	376	(3)
Examples	379	(2)
Notation. Extensions. Fundamental rules	381	(2)
Integral estimates and the mean value theorem	383	(2)
Integrals over Regions in three and more Dimensions	385	(1)
Space Differentiation. Mass and Density	386	(2)
Reduction of the Multiple Integral to Repeated Single Integrals	388	(10)
Integrals over a rectangle	388	(2)
Change of order of integration. Differentiation under the integral sign	390	(2)
Reduction of double integrals to single integrals for more general regions	392	(5)
Extension of the results to regions in several dimensions	397	(1)
Transformation of Multiple Integrals	398	(8)
Transformation of integrals in the plane	398	(5)
Regions of more than two dimensions	403	(3)
Improper Multiple Integrals	406	(11)
Improper integrals of functions over bounded sets	407	(4)
Proof of the general convergence theorem for improper integrals	411	(3)
Integrals over unbounded regions	414	(3)
Geometrical Applications	417	(14)
Elementary calculation of volumes	417	(2)
General remarks on the calculation of volumes. Solids of revolution. Volumes in spherical coordinates	419	(2)
Area of a curved surface	421	(10)
Physical Applications	431	(14)
Moments and center of mass	431	(2)
Moments of inertia	433	(3)
The compound pendulum	436	(2)
Potential of attracting masses	438	(7)
Multiple Integrals in Curvilinear Coordinates	445	(8)
Resolution of multiple integrals	445	(3)
Application to areas swept out by moving curves and volumes swept out by moving surfaces. Guldin's formula. The polar planimeter	448	(5)
Volumes and Surface Areas in Any Number of Dimensions	453	(9)
Surface areas and surface integrals in more than three dimensions	453	(2)
Area and volume of the n-dimensional sphere	455	(4)
Generalizations. Parametric Representations	459	(3)
Improper Single Integrals as Functions of a Parameter	462	(14)
Uniform convergence. Continuous dependence on the parameter	462	(4)
Integration and differentiation of improper integrals with respect to a parameter	466	(3)
Examples	469	(4)
Evaluation of Fresnel's integrals	473	(3)
The Fourier Integral	476	(21)
Introduction	476	(3)
Examples	479	(2)
Proof of Fourier's integral theorem	481	(4)
Rate of convergence in Fourier's integral theorem	485	(3)
Parseval's identity for Fourier transforms	488	(2)
The Fourier transformation for functions to several variables	490	(7)
The Eulerian Integrals (Gamma Function)	497	(18)
Definition and functional equation	497	(2)
Convex functions. Proof of Bohr and Mollerup's theorem	499	(4)
The infinite products for the gamma function	503	(4)
The nextensio theorem	507	(1)
The beta function	508	(3)
Differentiation and integration of fractional order. Abel's integral equation	511	(4)
APPENDIX: DETAILED ANALYSIS OF THE PROCESS OF INTEGRATION
A.1 Area	515	(9)
a. Subdivisions of the plane and the corresponding inner and outer areas	515	(2)
b. Jordan-measurable sets and their areas	517	(2)
c. Basic properties of areas	519	(5)
A.2 Integrals of Functions of Several Variables	524	(10)
a. Definition of the integral of a function f(x, y)	524	(2)
b. Integrability of continuous functions and integrals over sets	526	(2)
c. Basic rules for multiple integrals	528	(3)
d. Reduction of multiple integrals to repeated single integrals	531	(3)
A.3 Transformation of Areas and Integrals	534	(6)
a. Mappings of sets	534	(5)
b. Transformation of multiple integrals	539	(1)
A.4 Note on the Definition of the Area of a Curved Surface	540	(3)
Relations Between Surface and Volume Integrals
Connection Between Line Integrals and Double Integrals in the Plane (The Integral Theorems of Gauss, Stokes, and Green)	543	(8)
Vector Form of the Divergence Theorem. Stokes's Theorem	551	(5)
Formula for Integration by Parts in Two Dimensions. Green's Theorem	556	(2)
The Divergence Theorem Applied to the Transformation of Double Integrals	558	(7)
The case of 1-1 mappings	558	(3)
Transformation of integrals and degree of mapping	561	(4)
Area Differentiation. Transformation of Δu to Polar Coordinates	565	(4)
Interpretation of the Formulae of Gauss and Stokes by Two-Dimensional Flows	569	(6)
Orientation of Surfaces	575	(14)
Orientation of two-dimensional surfaces in three-space	575	(12)
Orientation of curves on oriented surfaces	587	(2)
Integrals of Differential Forms and of Scalars over Surfaces	589	(8)
Double integrals over oriented plane regions	589	(3)
Surface integrals of second-order differential forms	592	(2)
Relation between integrals of differential forms over oriented surfaces to integrals of scalars over unoriented surfaces	594	(3)
Gauss's and Green's Theorems in Space	597	(14)
Gauss's theorem	597	(5)
Application of Gauss's theorem to fluid flow	602	(3)
Gauss's theorem applied to space forces and surface forces	605	(2)
Integration by parts and Green's theorem in three dimensions	607	(1)
Application of Green's theorem to the transformation of Δ U to spherical coordinates	608	(3)
Stokes's Theorem in Space	611	(11)
Statement and proof of the theorem	611	(4)
Interpretation of Stokes's theorem	615	(7)
Integral Identities in Higher Dimensions	622	(2)
APPENDIX: GENERAL THEORY OF SURFACES AND OF SURFACE INTEGALS
A.1 Surfaces and Surface Integrals in Three dimensions	624	(13)
a. Elementary surfaces	624	(3)
b. Integral of a function over an elementary surface	627	(2)
c. Oriented elementary surfaces	629	(2)
d. Simple surfaces	631	(3)
e. Partitions of unity and integrals over simple surfaces	634	(3)
A.2 The Divergence Theorem	637	(5)
a. Statement of the theorem and its invariance	637	(2)
b. Proof of the theorem	639	(3)
A.3 Stokes's Theorem	642	(3)
A.4 Surfaces and Surface Integrals in Euclidean Spaces of Higher Dimensions	645	(6)
a. Elementary surfaces	645	(2)
b. Integral of a differential form over an oriented elementary surface	647	(1)
c. Simple m-dimensional surfaces	648	(3)
A.5 Integrals over Simple Surfaces, Gauss's Divergence Theorem, and the General Stokes Formula in Higher Dimensions	651	(3)
Differential Equations
The Differential Equations for the Motion of a Particle in Three Dimensions	654	(24)
The equations of motion	654	(2)
The principle of conservation of energy	656	(3)
Equilibrium. Stability	659	(2)
Small oscillations about a position of equilibrium	661	(4)
Planetary motion	665	(7)
Boundary value problems. The loaded cable and the loaded beam	672	(6)
The General Linear Differential Equation of the First Order	678	(5)
Separation of variables	678	(2)
The linear first-order equation	680	(3)
Linear Differential Equations of Higher Order	683	(14)
Principle of superposition. General solutions	683	(5)
Homogeneous differential equations of the second second order	688	(3)
The non-homogeneous differential equations. Method of variation of parameters	691	(6)
General Differential Equations of the First Order	697	(12)
Geometrical interpretation	697	(2)
The differential equation of a family of curves. Singular solutions. Orthogonal trajectories	699	(3)
Theorem of the existence and uniqueness of the solution	702	(7)
Systems of Differential Equations and Differential Equations of Higher Order	709	(2)
Integration by the Method of Undermined Coefficients	711	(2)
The Potential of Attracting Charges and Laplace's Equation	713	(14)
Potentials of mass distributions	713	(5)
The differential equation of the potential	718	(1)
Uniform double layers	719	(3)
The mean value theorem	722	(2)
Boundary value problem for the circle. Poisson's integral	724	(3)
Further Examples of Partial Differential Equations from Mathematical Physics	727	(10)
The wave equation in one dimension	727	(1)
The wave equation in three-dimensional space	728	(3)
Maxwell's equations in free space	731	(6)
Calculus of Variations
Functions and Their Extrema	737	(4)
Necessary conditions for Extreme Values of a Functional	741	(12)
Vanishing of the first variation	741	(2)
Deduction of Euler's differential equation	743	(4)
Proofs of the fundamental lemmas	747	(1)
Solution of Euler's differential equation in special cases. Examples	748	(4)
Identical vanishing of Euler's expression	752	(1)
Generalizations	753	(9)
Integrals with more than one argument function	753	(2)
Examples	755	(2)
Hamilton's principle. Lagrange's equations	757	(2)
Integrals involving higher derivatives	759	(1)
Several independent variables	760	(2)
Problems Involving Subsidiary Conditions. Lagrange Multipliers	762	(7)
Ordinary subsidiary conditions	762	(3)
Other types of subsidiary conditions	765	(4)
Functions of a Complex Variable
Complex Functions Represented by Power Series	769	(9)
Limits and infinite series with complex terms	769	(3)
Power series	772	(1)
Differentiation and integration of power series	773	(3)
Examples of power series	776	(2)
Foundations of the General Theory of Functions of a Complex Variable	778	(9)
The postulate of differentiability	778	(4)
The simplest operations of the differential calculus	782	(3)
Conformal transformation. Inverse functions	785	(2)
The Integration of Analytic Functions	787	(10)
Definition of the integral	787	(2)
Cauchy's theorem	789	(3)
Applications. The logarithm, the exponential function, and the general power function	792	(5)
Cauchy's Formula and Its Applications	797	(10)
Cauchy's formula	797	(2)
Expansion of analytic functions in power series	799	(3)
The theory of functions and potential theory	802	(1)
The converse of Cauchy's theorem	803	(1)
Zeros, poles, and residues of an analytic function	803	(4)
Applications to Complex Integration (Contour Integration)	807	(7)
Proof of the formula (8.22)	807	(1)
Proof of the formula (8.22)	808	(1)
Application of the theorem of residues to the integration of rational functions	809	(3)
The theorem of residues and linear differential equations with constant coefficients	812	(2)
Many-Valued Functions and Analytic Extension	814	(127)
List of Biographical Dates	941	(2)
Index	943