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This is the second book of our series “Higher Mathematics’. The series consists of the following three books: (1) Fundamentals of Linear Algebra and Analytical Geometry; (2) Differential and Integral Calculus; (3) Differential Equations. Multiple Integrals. Series. Functions of a Complex Variable. The book presupposes no special preparation beyond the ability to perform ordinary algebraic operations and some elementary knowledge of geometry acquired at high school. Though the text (Chapters 1 to 7) centres on calculus of one variable, analytic geometry and calculus of several variables are also developed (in Chapter 8). The closing chapter (9) deals with series. We think that this book and the first book of the series should be studied parallelly. Therefore, both books are provided with relevant cross-references. Our series “Higher Mathematics” will be completed with a book of problems for all the three books. The latter will appear in 1983. Translated from the Russian by Leonid Levant Volume 1 here CONTENTS Preface Chapter 1. Introduction Sec. 1.1 The Subject of Mathematics Variable and Constant Quantities Sets Sec. 1.2 Operations with Sets Sec. 1.3 Symbols of Mathematical Logic Sec. 1.4 Real Numbers Sec. 1.5 Defining an Equality and Inequality Sec. 1.6 Defining Arithmetic Operations Sec. 1.7 Basic Properties of Real Numbers Sec. 1.8 Axiomatic Approach to the Theory of Real Numbers Sec. 1.9 Inequalities for Absolute Values Sec. 1.10 Intervals Bounded Sets Sec. 1.11 Countable Sets Countability of the Set of Rational Numbers Uncountability of the Set of Real Numbers Chapter 2. Limit of Sequence Sec. 2.1 Concept of Limit of Sequence Sec. 2.2 Arithmetic Operations with Variables Having a Limit Sec. 2.3 Infinitesimals and Infinities Sec. 2.4 Evaluation of Indeterminate Forms Sec. 2.5 Monotone Sequences Sec. 2.6 The Number e Sec. 2.7 The Principle of Nested Intervals Sec. 2.8 Supremum and Infimum of a Set Sec. 2.9 The Bolzano-Weierstrass Theorem Sec. 2.10 Limit Superior and Limit Inferior Sec. 2.11 Cauchy Condition for Convergence of a Sequence Sec. 2.12 Completeness and Continuity of the Set of Real Numbers Chapter 3. Function Limit of Function Sec. 3.1 Function Sec. 3.2 Limit of a Function Sec. 3.3 Continuity of a Function Sec. 3.4 Discontinuities of the First and Second Kind Sec. 3.5 Functions Continuous on a Closed Interval Sec. 3.6 Inverse of a Continuous Function Sec. 3.7 Uniform Continuity of a Function Sec. 3.8 Elementary Functions Sec. 3.9 Remarkable Limits Sec. 3.10 Order of a Variable Equivalence Chapter 4. Differential Calculus Functions of One Variable Sec. 4.1 The Derivative Defined Sec. 4.2 Geometrical Meaning of the Derivative Sec. 4.3 Derivatives of Elementary Functions Sec. 4.4 Derivative of Composite Function Sec. 4.5 Derivative of Inverse Function Sec. 4.6 Derivatives of Elementary Functions Continued Sec. 4.7 Differential of a Function Sec. 4.8 Another Definition of Tangent Sec. 4.9 Derivatives of Higher Orders Sec. 4.10 Differentials of Higher Orders The Invariant Property of the First-order Differential Sec. 4.11 Differentiation of Functions Represented Parametrically Sec. 4.12 Mean-value Theorems for Derivatives Sec. 4.13 Evaluation of Indeterminate Forms Sec. 4.14 Taylor’s Formula Sec. 4.15 Taylor’s Series Sec. 4.16 Taylor’s Formulas and Series for Most Important Elementary Functions Sec. 4.17 Local Extremum of a Function Sec. 4.18 Extreme Values of a Function on a Closed Interval Sec. 4.19 Convexity of a Curve Points of Inflection Sec. 4.20 Asymptotes Sec. 4.21 A Continuous and Smooth Curve Sec. 4.22 General Scheme for Constructing the Graph of a Function Sec. 4.23 Vector Function The Vectors of the Tangent and Normal Chapter 5. Indefinite Integrals Sec. 5.1 Indefinite Integral Basic Table of Integrals Sec. 5.2 Methods of Integration Sec. 5.3 Complex Numbers Sec. 5.4 Theory of Polynomial of nth Degree Sec. 5.5 Real Polynomial of nth Degree Sec. 5.6 Integrating Rational Expressions Sec. 5.7 Integration of Irrational Functions Chapter 6. The Definite Integral Sec. 6.1 Some Problems of Geometry and Physics Leading to the Notion of the Definite Integral and Its Definition Sec. 6.2 Properties of Definite Integrals Sec. 6.3 Integral as Function of Its Upper Limit Sec. 6.4 The Newton-Leibniz Formula Sec. 6.5 Taylor’s Formula with the Remainder in the Integral Form Sec. 6.6 Darboux Sums Existence Conditions for the Integral Sec. 6.7 Integrability of Continuous and Monotone Functions Sec. 6.8 Improper Integrals Sec. 6.9 Improper Integrals of Nonnegative Functions Sec. 6.10 Improper Integrals Integrated by Parts Sec. 6.11 Improper Integrals with Singularities at Several Points Chapter 7. Some Applications of Integrals Approximate Methods Sec. 7.1 Area in Polar Coordinates Sec. 7.2 Volume of a Solid of Revolution Sec. 7.3 A Smooth Curve in Space Arc Length Sec. 7.4 Curvature and Radius of Curvature of a Curve Evolute and Evolvent Sec. 7.5 Area of a Surface of Revolution Sec. 7.6 Lagrange’s Interpolation Formula Sec. 7.7 Rectangle and Trapezoid Quadrature Formulas Sec. 7.8 Simpson’s Formula Chapter 8. Differential Calculus Functions of Several Variables Sec. 8.1 Preliminaries Sec. 8.2 Open Set Sec. 8.3 Limit of a Function Sec. 8.4 Continuous Functions Sec. 8.5 Partial Derivatives and Directional Derivative Sec. 8.6 Differentiable Functions Sec. 8.7 Tangent Plane Geometrical Meaning of Differential Sec. 8.8 Derivative of a Composite Function Directional Derivative Gradient Sec. 8.9 Differential of a Function Differentials of Higher Orders Sec. 8.10 Taylor’s Formula Sec. 8.11 Closed Sets Sec. 8.12 Functions Continuous on Bounded Closed Sets Sec. 8.13 Extrema Sec. 8.14 Finding the Greatest and the Least Values of Functions Sec. 8.15 Existence Theorem for an Implicit Function Sec. 8.16 Tangent Plane and Normal Sec. 8.17 Systems of Functions Represented Implicitly Sec. 8.18 Mappings Sec. 8.19 Conditional Relative Extremum Chapter 9. Series Sec. 9.1 The Notion of a Series Sec. 9.2 Improper Integral and Series Sec. 9.3 Operations with Series Sec. 9.4 Series with Nonnegative Terms Sec. 9.5 Leibniz’ Series Sec. 9.6 Absolutely Convergent Series Sec. 9.7 Conditionally and Unconditionally Convergent Series with Real Terms Sec. 9.8 Sequences and Series of Functions Uniform Convergence Sec. 9.9 Integration and Differentiation of Uniformly Convergent Series Sec. 9.10 Multiplication of Absolutely Convergent Series Sec. 9.11 Power Series Sec. 9.12 Differentiation and Integration of Power Series Sec. 9.13 The Functions e^z sin z and cos z of Complex Variable z Sec. 9.14 Series in Approximate Computations Subject Index

This is the second book of our series “Higher Mathematics’. The series consists of the following three books: (1) Fundamentals of Linear Algebra and Analytical Geometry; (2) Differential and Integral Calculus; (3) Differential Equations. Multiple Integrals. Series. Functions of a Complex Variable. The book presupposes no special preparation beyond the ability to perform ordinary algebraic operations and some elementary knowledge of geometry acquired at high school. Though the text (Chapters 1 to 7) centres on calculus of one variable, analytic geometry and calculus of several variables are also developed (in Chapter 8). The closing chapter (9) deals with series. We think that this book and the first book of the series should be studied parallelly. Therefore, both books are provided with relevant cross-references. Our series “Higher Mathematics” will be completed with a book of problems for all the three books. The latter will appear in 1983. Translated from the Russian by Leonid Levant Volume 1 here CONTENTS Preface Chapter 1. Introduction Sec. 1.1 The Subject of Mathematics Variable and Constant Quantities Sets Sec. 1.2 Operations with Sets Sec. 1.3 Symbols of Mathematical Logic Sec. 1.4 Real Numbers Sec. 1.5 Defining an Equality and Inequality Sec. 1.6 Defining Arithmetic Operations Sec. 1.7 Basic Properties of Real Numbers Sec. 1.8 Axiomatic Approach to the Theory of Real Numbers Sec. 1.9 Inequalities for Absolute Values Sec. 1.10 Intervals Bounded Sets Sec. 1.11 Countable Sets Countability of the Set of Rational Numbers Uncountability of the Set of Real Numbers Chapter 2. Limit of Sequence Sec. 2.1 Concept of Limit of Sequence Sec. 2.2 Arithmetic Operations with Variables Having a Limit Sec. 2.3 Infinitesimals and Infinities Sec. 2.4 Evaluation of Indeterminate Forms Sec. 2.5 Monotone Sequences Sec. 2.6 The Number e Sec. 2.7 The Principle of Nested Intervals Sec. 2.8 Supremum and Infimum of a Set Sec. 2.9 The Bolzano-Weierstrass Theorem Sec. 2.10 Limit Superior and Limit Inferior Sec. 2.11 Cauchy Condition for Convergence of a Sequence Sec. 2.12 Completeness and Continuity of the Set of Real Numbers Chapter 3. Function Limit of Function Sec. 3.1 Function Sec. 3.2 Limit of a Function Sec. 3.3 Continuity of a Function Sec. 3.4 Discontinuities of the First and Second Kind Sec. 3.5 Functions Continuous on a Closed Interval Sec. 3.6 Inverse of a Continuous Function Sec. 3.7 Uniform Continuity of a Function Sec. 3.8 Elementary Functions Sec. 3.9 Remarkable Limits Sec. 3.10 Order of a Variable Equivalence Chapter 4. Differential Calculus Functions of One Variable Sec. 4.1 The Derivative Defined Sec. 4.2 Geometrical Meaning of the Derivative Sec. 4.3 Derivatives of Elementary Functions Sec. 4.4 Derivative of Composite Function Sec. 4.5 Derivative of Inverse Function Sec. 4.6 Derivatives of Elementary Functions Continued Sec. 4.7 Differential of a Function Sec. 4.8 Another Definition of Tangent Sec. 4.9 Derivatives of Higher Orders Sec. 4.10 Differentials of Higher Orders The Invariant Property of the First-order Differential Sec. 4.11 Differentiation of Functions Represented Parametrically Sec. 4.12 Mean-value Theorems for Derivatives Sec. 4.13 Evaluation of Indeterminate Forms Sec. 4.14 Taylor’s Formula Sec. 4.15 Taylor’s Series Sec. 4.16 Taylor’s Formulas and Series for Most Important Elementary Functions Sec. 4.17 Local Extremum of a Function Sec. 4.18 Extreme Values of a Function on a Closed Interval Sec. 4.19 Convexity of a Curve Points of Inflection Sec. 4.20 Asymptotes Sec. 4.21 A Continuous and Smooth Curve Sec. 4.22 General Scheme for Constructing the Graph of a Function Sec. 4.23 Vector Function The Vectors of the Tangent and Normal Chapter 5. Indefinite Integrals Sec. 5.1 Indefinite Integral Basic Table of Integrals Sec. 5.2 Methods of Integration Sec. 5.3 Complex Numbers Sec. 5.4 Theory of Polynomial of nth Degree Sec. 5.5 Real Polynomial of nth Degree Sec. 5.6 Integrating Rational Expressions Sec. 5.7 Integration of Irrational Functions Chapter 6. The Definite Integral Sec. 6.1 Some Problems of Geometry and Physics Leading to the Notion of the Definite Integral and Its Definition Sec. 6.2 Properties of Definite Integrals Sec. 6.3 Integral as Function of Its Upper Limit Sec. 6.4 The Newton-Leibniz Formula Sec. 6.5 Taylor’s Formula with the Remainder in the Integral Form Sec. 6.6 Darboux Sums Existence Conditions for the Integral Sec. 6.7 Integrability of Continuous and Monotone Functions Sec. 6.8 Improper Integrals Sec. 6.9 Improper Integrals of Nonnegative Functions Sec. 6.10 Improper Integrals Integrated by Parts Sec. 6.11 Improper Integrals with Singularities at Several Points Chapter 7. Some Applications of Integrals Approximate Methods Sec. 7.1 Area in Polar Coordinates Sec. 7.2 Volume of a Solid of Revolution Sec. 7.3 A Smooth Curve in Space Arc Length Sec. 7.4 Curvature and Radius of Curvature of a Curve Evolute and Evolvent Sec. 7.5 Area of a Surface of Revolution Sec. 7.6 Lagrange’s Interpolation Formula Sec. 7.7 Rectangle and Trapezoid Quadrature Formulas Sec. 7.8 Simpson’s Formula Chapter 8. Differential Calculus Functions of Several Variables Sec. 8.1 Preliminaries Sec. 8.2 Open Set Sec. 8.3 Limit of a Function Sec. 8.4 Continuous Functions Sec. 8.5 Partial Derivatives and Directional Derivative Sec. 8.6 Differentiable Functions Sec. 8.7 Tangent Plane Geometrical Meaning of Differential Sec. 8.8 Derivative of a Composite Function Directional Derivative Gradient Sec. 8.9 Differential of a Function Differentials of Higher Orders Sec. 8.10 Taylor’s Formula Sec. 8.11 Closed Sets Sec. 8.12 Functions Continuous on Bounded Closed Sets Sec. 8.13 Extrema Sec. 8.14 Finding the Greatest and the Least Values of Functions Sec. 8.15 Existence Theorem for an Implicit Function Sec. 8.16 Tangent Plane and Normal Sec. 8.17 Systems of Functions Represented Implicitly Sec. 8.18 Mappings Sec. 8.19 Conditional Relative Extremum Chapter 9. Series Sec. 9.1 The Notion of a Series Sec. 9.2 Improper Integral and Series Sec. 9.3 Operations with Series Sec. 9.4 Series with Nonnegative Terms Sec. 9.5 Leibniz’ Series Sec. 9.6 Absolutely Convergent Series Sec. 9.7 Conditionally and Unconditionally Convergent Series with Real Terms Sec. 9.8 Sequences and Series of Functions Uniform Convergence Sec. 9.9 Integration and Differentiation of Uniformly Convergent Series Sec. 9.10 Multiplication of Absolutely Convergent Series Sec. 9.11 Power Series Sec. 9.12 Differentiation and Integration of Power Series Sec. 9.13 The Functions e^z sin z and cos z of Complex Variable z Sec. 9.14 Series in Approximate Computations Subject Index

Book digitized by Google from the library of the University of Michigan and uploaded to the Internet Archive by user tpb.

480 pages 23 cm

xviii, 481 p. 22 cm

xviii p., 1 \U+fffd\., 343 p. 21 cm

xviii, 491 p. 22 cm

xviii, 491 p. 22 cm

xi, 295 p. 22 cm

Book digitized by Google from the library of the University of Michigan and uploaded to the Internet Archive by user tpb.

Book digitized by Google and uploaded to the Internet Archive by user tpb.

Book digitized by Google from the library of Harvard University and uploaded to the Internet Archive by user tpb.

Book digitized by Google from the library of Harvard University and uploaded to the Internet Archive by user tpb.

283 p. 21 cm

283 p. 21 cm

14

Source: Digital Library of India Scanning Centre: Allama Iqbal Library, University of Kashmir Source Library: Amar Singh College, Srinagar Date Accessioned: 7/18/2015 0:38 The Digital Library of India was a project under the auspices of the Government of India.

Source: Digital Library of India Scanning Centre: Allama Iqbal Library, University of Kashmir Source Library: Amar Singh College, Srinagar Date Accessioned: 7/18/2015 0:38 The Digital Library of India was a project under the auspices of the Government of India.

First issued in 25 parts, July 15, 1836, to June 1, 1842

283 p. : 21 cm

283 p. : 21 cm

283 p. : 21 cm

Book digitized by Google and uploaded to the Internet Archive by user tpb.

ix, 307 p. 20 cm

xi, 256 p. 22 cm

641 p. :

Book digitized by Google from the library of the University of Michigan and uploaded to the Internet Archive by user tpb.

Book digitized by Google from the library of Harvard University and uploaded to the Internet Archive by user tpb.

Book digitized by Google from the library of Harvard University and uploaded to the Internet Archive by user tpb.

Book digitized by Google from the library of the University of Michigan and uploaded to the Internet Archive by user tpb.

Book digitized by Google from the library of the University of Michigan and uploaded to the Internet Archive by user tpb.

Book digitized by Google and uploaded to the Internet Archive by user tpb.

Book digitized by Google from the library of Harvard University and uploaded to the Internet Archive by user tpb.

Book digitized by Google from the library of Harvard University and uploaded to the Internet Archive by user tpb.

viii, [9]-309 p. 20 cm

xiv, 395 p. : 19 cm

501 p. ; 22 cm

Differential and Integral Calculus by N. Piskunov This text is designed as a course of mathematics for higher technical schools. It contains many worked examples that illustrate the theoretical material and serve as models for solving problems. The first two chapters "Number. Variable. Function" and "Limit. Continuity of a Function" have been made as short as possible. Some of the questions that are usually discussed in these chapters have been put in the third and subsequent chapters without loss of continuity. This has made it possible to take up very early the basic concept of differential calculusâthe derivativeâ which is required in the study of technical subjects. Experience has shown this arrangement of the material to be the best and most convenient for the student. A large number of problems have been included, many of which illustrate the interrelationships of mathematics and other disciplines. The problems are specially selected (and in sufficient number) for each section of the course thus helping the student to master the theoretical material. To a large extent, this makes the use of a separate book of problems unnecessary and extends the usefulness of this text as a course of mathematics for self-instruction. The book was translated from the Russian by G. Yankovsky and was published by Mir in 1969. Subsequently it was also republished as a single and two volume format. You can get the book here. Contents Preface 11 Chapter I NUMBER. VARIABLE. FUNCTION 1. Real Numbers. Real Numbers as Points on a Number Scale 13 2. The Absolute Value of a Real Number 14 3. Variables and Constants 16 4. The Range of a Variable 16 5. Ordered Variables. Increasing and Decreasing Variables. Bounded Variables 18 6. Function 19 7. Ways of Representing Functions 20 8. Basic Elementary Functions. Elementary Functions 22 9. Algebraic Functions " 26 10. Polar Coordinate System 28 Exercises on Chapter 7 30 Chapter II. LIMIT. CONTINUITY OF A FUNCTION 1. The Limit of a Variable. An Infinitely Large Variable 32 2. The Limit of a Function 35 3. A Function that Approaches Infinity Bounded Functions 38 4. Infinitesimals and Their Basic Properties 42 5. Basic Theorems on Limits 45 6. The Limit of the Function $latex \frac(\sin x)(x) \, x \to 0$. 50 7. The Number e 51 8. Natural Logarithms 56 9. Continuity of Functions 57 10. Certain Properties of Continuous Functions 61 11. Comparing Infinitesimals 63 Exercises on Chapter 77 66 Chapter III. DERIVATIVE AND DIFFERENTIAL 1. Velocity of Motion 69 2. Definition of Derivative 71 3. Geometric Meaning of the Derivative 73 4. Differentiability of Functions 74 5. Finding the Derivatives of Elementary Functions. The Derivative of the Function $y = x^n$ Where n Is Positive and Integral 76 6. Derivatives of the Functions $y=s\nx\ y = cosx$ . 78 7. Derivatives of: a Constant, the Product of a Constant by a Function, a Sum, a Product, and a Quotient 80 8. The Derivative of a Logarithmic Function 84 9. The Derivative of a Composite Function . 85 10. Derivatives of the Functions y = ianx, y = coixt y=\n\x\ 83 11. An Implicit Function and Its Differentiation 89 12. Derivatives of a Power Function for an Arbitrary Real Exponent, of an Exponential Function, and a Composite Exponential Function . . 91 13. An Inverse Function and Its Differentiation 94 14. Inverse Trigonometric Functions and Their Differentiation 98 15. Table of Basic Differentiation Formulas 102 16. Parametric Representation of a Function 103 17. The Equations of Certain Curves in Parametric Form 105 18. The Derivative of a Function Represented Parametrically 108 19. Hyperbolic Functions 110 20. The Differential 113 21. The Geometric Significance of the Differential 117 22. Derivatives of Different Orders 118 23. Differentials of Various Orders 121 24. Different-Order Derivatives of Implicit Functions and of Functions Represented Parametrically 122 25. The Mechanical Significance of the Second Derivative 124 26. The Equations of a Tangent and of a Normal. The Lengths of the Subtangent and the Subnormal 126 27. The Geometric Significance of the Derivative of the Radius Vector with Respect to the Polar Angle 129 Exercises on Chapter III 130 Chapter IV. SOME THEOREMS ON DIFFERENTIABLE FUNCTIONS 1. A Theorem on the Roots of a Derivative (Rolle's Theorem) .... 140 2. A Theorem on Finite Increments (Lagrange's Theorem) 142 3. A Theorem on the Ratio of the Increments of Two Functions (Cauchy's Theorem) - 143 4. The Limit of a Ratio of Two Infinitesimals ( Evaluation of Indeterminate Forms of the Type -rr- J 144 5. The Limit of a Ratio of Two Infinitely Large Quantities (Evaluation 00 \ of Indeterminate Forms of the Type â 147 6. Taylor's Formula 152 7. Expansion of the Functions e*, sin*, and cos x in a Taylor Series . 156 Exercises on Chapter IV 159 Chapter V. INVESTIGATING THE BEHAVIOUR OF FUNCTIONS 1. Statement of the Problem 162 2. Increase and Decrease of a Function 163 3. Maxima and Minima of Functions 164 4. Testing a Differentiable Function for Maximum and Minimum with a First Derivative 171 5. Testing a Function for Maximum and Minimum with a Second Derivative 174 6. Maxima and Minima of a Function on an Interval 178 7. Applying the Theory of Maxima and Minima of Functions to the Solution of Problems 179 8. Testing a Function for Maximum and Minimum by Means of Taylor's Formula 181 9. Convexity and Concavity of a Curve. Points of Inflection 183 10. Asymptotes ' 189 11. General Plan for Investigating Functions and Constructing Graphs 194 12. Investigating Curves Represented Parametrically 199 Exercises on Chapter V . 203 Chapter VI. THE CURVATURE OF A CURVE 1. The Length of an Arc and Its Derivative 208 2. Curvature 210 3. Calculation of Curvature 212 4. Calculation of the Curvature of a Line Represented Parametrically . . 215 5. Calculation of the Curvature of a Line Given by an Equation of Polar Coordinates 215 6. The Radius and Circle of Curvature. Centre of Curvature. Evolute and Involute 217 7. The Properties of an Evolute 221 8. Approximating the Real Roots of an Equation 225 Exercises on Chapter VI -. . . . 229 Chapter VII. COMPLEX NUMBERS. POLYNOMIALS 1. Complex Numbers. Basic Definitions 233 2. Basic Operations on Complex Numbers : 234 3. Powers and Roots of Complex Numbers 237 4. Exponential Function with Complex Exponent and Its Properties . . 240 5. Euler's Formula. The Exponential Form of a Complex Number . . . 243 6. Factoring a Polynomial . . . . 244 7. The Multiple Roots of a Polynomial 247. 8. Factorisation of a Polynomial in the Case of Complex Roots .... 248 9. Interpolation. Lagrange's Interpolation Formula 250 10. On the Best Approximation of Functions by Polynomials. Chebyshev's Theory . . . . . 252 Exercises on Chapter VII 253 Chapter VIII. FUNCTIONS OF SEVERAL VARIABLES 1. Definition of a Function of Several Variables 255 2. Geometric Representation of a Function of Two Variables ..... 25§ 3. Partial and Total Increment of a Function 259 4. Continuity of a Function of Several Variables 260 5. Partial Derivatives of a Function of Several Variables 263 6. The Geometric Interpretation of the Partial Derivatives of a Function of Two Variables 264 7. Total Increment and Total Differentials 265 8. Approximation by Total Differentials 268 9. Error Approximation by Differentials 270 10. The Derivative of a Composite Function. The Total Derivative . . . 273 11. The Derivative of a Function Defined Implicitly 276 12. Partial Derivatives of Different Orders 279 13. Level Surfaces .283 14. Directional Derivatives . 284 15. Gradient 286 16. Taylor's Formula for a Function of Two Variables 290 17. Maximum and-Minimum of a Function of Several Variables .... 292 18. Maximum and Minimum of a Function of Several Variables Related by Given Equations (Conditional Maxima and Minima) 300 19. Singular Points of a Curve 305 Exercises on Chapter VIII 310 Chapter IX. APPLICATIONS OF DIFFERENTIAL CALCULUS TO SOLID GEOMETRY 1. The Equations of a Curve in Space 314 2. The Limit and Derivative of the Vector Function of a Scalar Argument. The Equation of a Tangent to a Curve. The Equation of a Normal Plane 317 3. Rules for Differentiating Vectors (Vector Functions) 322 4. The First and Second Derivatives of a Vector with Respect to the Arc Length. The Curvature of a Curve. The Principal Normal .... 324 5. Osculating Plane. Binormal. Torsion 331 6. A Tangent Plane and Normal to a Surface 336 Exercises on Chapter IX 340 Chapter X, INDEFINITE INTEGRALS 1. Antiderivative and the Indefinite Integral 342 2. Table of Integrals 344 3. Some Properties of an Indefinite Integral 346 4. Integration by Substitution (Change of Variable) 348 5. Integrals of Functions Containing a Quadratic Trinomial 351 6. Integration by Parts 354 7. Rational Fractions. Partial Rational Fractions and Their Integration 357 8. Decomposition of a Rational Fraction into Partial Fractions .... 361 9. Integration of Rational Fractions 365 10. Ostrogradsky's Method 368 11. Integrals of Irrational Functions 371 12. Integrals of the Form [R(x> V'ax2+bx + c)dx . 372 13. Integration of Binomial Differentials 375 14. Integration of Certain Classes of Trigonometric Functions 378 15. Integration of Certain Irrational Functions by Means of Trigonometric Substitutions 383 16. Functions Whose Integrals Cannot Be Expressed in Terms of Elementary Functions 385 Exercises on Chapter X 386 Chapter XI. THE DEFINITE INTEGRAL 1. Statement of the Problem. The Lower and Upper Integral Sums . . . 396 2. The Definite Integral 398 3. Basic Properties o? the Definite Integral . 404 4. Evaluating a Definite Integral. Newton-Leibniz Formula 407 5. Changing the Variable in the Definite Integral 412 6. Integration by Parts 413 7. Improper Integrals 416 8. Approximating Definite Integrals 424 9. Chebyshev's Formula 430 10. Integrals Dependent on a Parameter 435 Exercises on Chapter XI 438 Chapter XII. GEOMETRIC AND MECHANICAL APPLICATIONS OF THE DEFINITE INTEGRAL 1. Computing Areas in Rectangular Coordinates 442 2. The Area of a Curvilinear Sector in Polar Coordinates ' . 445 3. The Arc Length of a Curve . 447 4. Computing the Volume of a Solid from the Areas of Parallel Sections (Volumes by Slicing) 453 5. The Volume of a Solid of Revolution 455 6. The Surface of a Solid of Revolution 455 7. Computing Work by the Definite Integral 457 8. Coordinates of the Centre of Gravity 459 Exercises on Chapter XII 462 Chapter XIII. DIFFERENTIAL EQUATIONS 1. Statement of the Problem. The Equation of Motion of a Body with Resistance of the Medium Proportional to the Velocity. The Equation of a Catenary 469 2. Definitions \ . . , 472 3. First-Order Differential Equations (General Notions) . 473 4. Equations with Separated and Separable Variables. The Problem of the Disintegration of Radium 478 5. Homogeneous First-Order Equations 482 6. Equations Reducible to Homogeneous Equations 484 7. First-Order Linear Equations 487 8. Bernoulli's Equation 490 9. Exact Differential Equations 492 10. Integrating Factor 495 11. The Envelope of a Family of Curves 497 12. Singular Solutions of a First-Order Differential Equation 504 13. Clairaut's Equation 505 14. Lagrange's Equation 507 15. Orthogonal and Isogonal Trajectories 509 16. Higher-Order Differential Equations (Fundamentals) 514 17. An Equation of the Form y = f(x) 516 18. Some Types of Second-Order Differential Equations Reducible to First-Order Equations . . 518 19. Graphical Method of Integrating Second-Order Differential Equations 527 20. Homogeneous Linear Equations. Definitions and General Properties 528 21. Second-Order Homogeneous Linear Equations with Constant Coefficients ( 535 22. Homogeneous Linear Equations of the Aith Order with Constant Coefficients 539 23. Nonhomogeneous Second-Order Linear Equations 541 24. Nonhomogeneous Second-Order Linear Equations with Constant Coefficients 545 25. Higher-Order Nonhomogeneous Linear Equations 551 26. The Differential -Equation of Mechanical Vibrations 555 27. Free Oscillations' 557 28. Forced Oscillations 559 29. Systems of Ordinary Differential Equations 563 30. Systems of Linear Differential Equations with Constant Coefficients 569 31. On Lyapunov's Theory of Stability 576 32. Euler's Method of Approximate Solution of First-Order Differential Equations . 581 33. A Difference Method for Approximate Solution of Differential Equations Based on Taylor's Formula. Adams Method 584 34. An Approximate Method for Integrating Systems of First-Order Differential Equations 591 Exercises on Chapter XIII 595 Chapter XIV. MULTIPLE INTEGRALS 1. Double Integrals . 608 2. Calculating Double Integrals 610 3. Calculating Double Integrals (Continued) 617 4. Calculating Areas and Volumes by Means of Double Integrals .... 623 5. The Double Integral in Polar Coordinates 626 6. Changing Variables in a Double Integral (General Case) 633 7. Computing the Area of a Surface 638 8. The Density of Distribution of Matter and the Double Integral . . . 642 9. The Moment of Inertia of the Area of a Plane Figure 643 10. The Coordinates of the Centre of Gravity of the Area of a Plane Figure 648 11. Triple Integrals 650 12. Evaluating a Triple Integral 651 13. Change of Variables in a Triple Integral 656 14. The Moment of Inertia and the Coordinates of the Centre of Gravity of a Solid . 660 15. Computing Integrals Dependent on a Parameter . 662 Exercises on Chapter XIV . . . 663 Chapter XV. LINE INTEGRALS AND SURFACE INTEGRALS 1. Line Integrals 670 2. Evaluating a Line Integral 673 3. Green's Formula 679 4. Conditions for a Line Integral Being Independent of the Path of Integration 681 5. Surface Integrals . 687 6. Evaluating Surface Integrals 689 7. Stokes' Formula 692 8. Ostrogradsky's Formula 697 9. The Hamiltonian Operator and Certain Applications of It 700 Exercises on Chapter XV 703 Chapter XVI. SERIES 1. Series. Sum of a Series 710 2. Necessary Condition for Convergence of a Series 713 3. Comparing Series with Positive Terms 716 4. D'Alembert's Test 718 5. Cauchy's Test 721 6. The Integral Test for Convergence of a Series .723 7. Alternating Series. Leibniz' Theorem 727 8. PIus-and-Minus Series. Absolute and Contitional Convergence .... 729 9. Functional Series 733 10. Majorised Series 734 11. The Continuity of the Sum of a Series 736 12. Integration and Differentiation of Series 739 13. Power Series. Interval of Convergence 742 14. Differentiation of Power Series 747 15. Series in Powers of â * âa 748 16. Taylor's Series and Maclaurin's Series .' 750 17. Examples of Expansion of Functions in Series 751 18. Euler's Formula 753 19. The Binomial Series 754 20. Expansion of the Function In (\+x) in a Power Series. Computing Logarithms 756 21. Integration by Use of Series (Calculating Definite Integrals) .... 758 22. Integrating Differential Equations by Means of Series ....... 760 23. Bessel's Equation 763 Exercises on Chapter XVI 768 Chapter XVII. FOURIER SERIES 1. Definition. Statement of the Problem 776 2. Expansions of Functions in Fourier Series 780 3. A Remark on the Expansion of a Periodic Function in a Fourier Series 785 4. Fourier Series for Even and Odd Functions 787 5. The Fourier Series for a Function with Period 2/ 789 6. On the Expansion of a Nonperiodic Function in a Fourier Series . . 791 7. Approximation by a Trigonometric Polynomial of a Function Represented in the Mean 792 8. The Dirichlet Integral 798 9. The Convergence of a Fourier Series at a Given Point . . . . . . .801 10. Certain Sufficient Conditions for the Convergence of a Fourier Series 802 11. Practical Harmonic Analysis 805 12. Fourier Integral 810 13. The Fourier Integral in Complex Form 810 Exercises on Chapter XVII 812 Chapter XVIII. EQUATIONS OF MATHEMATICAL PHYSICS 1. Basic Types of Equations of Mathematical Physics . 815 2. Derivation of the Equation of Oscillations of a String. Formulation of the Boundary-Value Problem. Derivation of Equations of Electric Oscillations in Wires : 816 3. Solution of the Equation of Oscillations of a String by the Method of Separation of Variables (The Fourier Method) 820 4. The Equation for Propagation of Heat in a Rod. Formulation of the Boundary-Value Problem 823 5. Heat Propagation in Space 825 6. Solution of the First Boundary-Value Problem for the Heat- Conductivity Equation by the Method of Finite Differences 829 7. Propagation of Heat in an Unbounded Rod 831 8. Problems That Reduce to Investigating Solutions of the Laplace Equation. Stating Boundary-Value Problems 836 9. The Laplace Equation in Cylindrical Coordinates. Solution of the Dirichlet Problem for a Ring with Constant Values of the Desired Function on the Inner and Outer Circumferences 841 10. The Solution of Dirichlet's Problem for a Circle 843 11. Solution of the Dirichlet Problem by the Method of Finite Differences 847 Exercises on Chapter XVIII 850 Chapter XIX. OPERATIONAL CALCULUS AND CERTAIN OF ITS APPLICATIONS 1. The Initial Function and Its Transform 854 2. Transforms of the Functions.oQ(t), sin/, cost 855 3. The Transform of a Function with Changed Scale of the Independent Variable. Transforms of the Functions sin at, cos at 856 4. The Linearity Property of a Transform 857 5. The Shift Theorem r 858 6. Transforms of the Functionse~at, sinh at, cosh at, e-aisinatt e~a/ cos at 858 7. Differentiation of Transforms 860 8. The Transforms of Derivatives 861 9. Table of Transforms ~. 862 10. An Auxiliary Equation for a Given Differential Equation 864 11. Decomposition Theorem . 867 12. Examples of Solutions of Differential Equations and Systems of Differential Equations by the Operational /Method 869 13. The Convolution Theorem 871 14. The Differential Equations of Mechanical Oscillations. The Differential Equations of Electric-Circuit Theory 873 15. Solution of the Differential Oscillation Equation 874 16. Investigating Free Oscillations 875 17. Investigating Mechanical and Electrical Oscillations in the Case of a Periodic External Force 876 18. Solving the Oscillation Equation in the Case of Resonance . . . . 878 19. The Delay Theorem 879 Exercises on Chapter XIX 880 Subject Index

xiii, 236 p. : 22 cm

Book digitized by Google and uploaded to the Internet Archive by user tpb.

Book digitized by Google from the library of Harvard University and uploaded to the Internet Archive by user tpb.

Book digitized by Google from the library of Harvard University and uploaded to the Internet Archive by user tpb.

Book digitized by Google from the library of Harvard University and uploaded to the Internet Archive by user tpb.

xii, [13]-344 p. 20 cm

xii, [13]-344 p. 20 cm

viii, 506 p. 23 cm

-number. variable. functions; limit. continuity of a function; derivative and differential; some theorems on differentiable functions; investigating the behaviour of functions; the curvature of a curve; complex numbers. polynomials; functions of several variables; applications of differential calculus to solid geometry; the indefinite integral; the definite integral; geometric and mechanical applications of the definite integral-

This text is designed as a course of mathematics for higher technical schools. It contains many worked examples that illustrate the theoretical material and serve as models for solving problems. The first two chapters "Number. Variable. Function" and "Limit.  Continuity of a Function" have been made as short as possible. Some of the questions that are usually discussed in these chapters have been put in the third and subsequent chapters without loss of continuity. This has made it possible to take up very early the basic concept of differential calculus—the derivative— which is required in the study of technical subjects. Experience has shown this arrangement of the material to be the best and most convenient for the student.

This text is designed as a course of mathematics for higher technical schools. It contains many worked examples that illustrate the theoretical material and serve as models for solving problems. The first two chapters "Number. Variable. Function" and "Limit.  Continuity of a Function" have been made as short as possible. Some of the questions that are usually discussed in these chapters have been put in the third and subsequent chapters without loss of continuity. This has made it possible to take up very early the basic concept of differential calculus—the derivative— which is required in the study of technical subjects. Experience has shown this arrangement of the material to be the best and most convenient for the student.

Book digitized by Google from the library of the University of Michigan and uploaded to the Internet Archive by user tpb.