Differential And Integral Calculus Piskunov Pdf Free Download
Piskunov's book is considered to be a classic.
This text is designed equally a course of mathematics for college technical schools. Information technology contains many worked examples that illustrate the theoretical cloth and serve as models for solving issues. The start ii chapters "Number. Variable. Function" and "Limit. Continuity of a Function" have been fabricated as brusk as possible. Some of the questions that are usually discussed in these chapters have been put in the tertiary and subsequent chapters without loss of continuity. This has made information technology possible to accept upwardly very early the basic concept of differential calculus—the derivative— which is required in the study of technical subjects. Experience has shown this system of the material to be the best and nearly convenient for the student.
A large number of issues 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 big extent, this makes the use of a separate book of problems unnecessary and extends the usefulness of this text equally a form of mathematics for self-teaching.
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The book was translated from the Russian past G. Yankovsky and was published by Mir in 1969. Subsequently it was also republished as a unmarried and ii volume format.
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Contents
Preface eleven
Chapter I NUMBER. VARIABLE. Role
1. Real Numbers. Existent Numbers as Points on a Number Scale 13
2. The Accented Value of a Existent Number 14
3. Variables and Constants sixteen
4. The Range of a Variable 16
5. Ordered Variables. Increasing and Decreasing Variables. Bounded Variables 18
6. Function 19
7. Means of Representing Functions 20
8. Bones Elementary Functions. Elementary Functions 22
nine. Algebraic Functions " 26
10. Polar Coordinate Arrangement 28
Exercises on Chapter 7 xxx
Affiliate 2. LIMIT. CONTINUITY OF A Part
ane. The Limit of a Variable. An Infinitely Large Variable 32
two. The Limit of a Function 35
three. A Function that Approaches Infinity Divisional Functions 38
4. Infinitesimals and Their Basic Properties 42
5. Basic Theorems on Limits 45
6. The Limit of the Function 
7. The Number east 51
8. Natural Logarithms 56
9. Continuity of Functions 57
10. Certain Backdrop of Continuous Functions 61
11. Comparison Infinitesimals 63
Exercises on Chapter 77 66
Affiliate Three. DERIVATIVE AND DIFFERENTIAL
ane. Velocity of Motility 69
ii. Definition of Derivative 71
iii. Geometric Pregnant of the Derivative 73
4. Differentiability of Functions 74
five. Finding the Derivatives of Uncomplicated Functions. The Derivative of the Function $y = x^north$ Where north Is Positive and Integral 76
half-dozen. Derivatives of the Functions $y=s\nx\ y = cosx$ . 78
vii. Derivatives of: a Constant, the Product of a Constant by a Function, a Sum, a Product, and a Quotient 80
viii. The Derivative of a Logarithmic Function 84
nine. The Derivative of a Blended 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 Ability Function for an Arbitrary Real Exponent, of an Exponential Part, and a Composite Exponential Function . . 91
thirteen. An Inverse Role and Its Differentiation 94
xiv. Inverse Trigonometric Functions and Their Differentiation 98
15. Tabular array of Basic Differentiation Formulas 102
16. Parametric Representation of a Function 103
17. The Equations of Sure Curves in Parametric Form 105
18. The Derivative of a Function Represented Parametrically 108
xix. Hyperbolic Functions 110
20. The Differential 113
21. The Geometric Significance of the Differential 117
22. Derivatives of Different Orders 118
23. Differentials of Diverse Orders 121
24. Different-Order Derivatives of Implicit Functions and of Functions Represented Parametrically 122
25. The Mechanical Significance of the 2nd 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 Bending 129
Exercises on Chapter 3 130
Affiliate IV. SOME THEOREMS ON DIFFERENTIABLE FUNCTIONS
one. A Theorem on the Roots of a Derivative (Rolle's Theorem) …. 140
ii. A Theorem on Finite Increments (Lagrange's Theorem) 142
3. A Theorem on the Ratio of the Increments of Two Functions (Cauchy'due south Theorem) – 143
4. The Limit of a Ratio of Ii Infinitesimals ( Evaluation of Indeterminate Forms of the Blazon -rr- J 144
five. The Limit of a Ratio of Ii Infinitely Large Quantities (Evaluation 00 \ of Indeterminate Forms of the Type — 147
6. Taylor's Formula 152
7. Expansion of the Functions e*, sin*, and cos x in a Taylor Serial . 156
Exercises on Chapter IV 159
Chapter Five. 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
five. 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 Ways of Taylor's Formula 181
ix. 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
ii. Curvature 210
3. Adding of Curvature 212
4. Calculation of the Curvature of a Line Represented Parametrically . . 215
v. Adding 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
vii. The Backdrop of an Evolute 221
eight. Approximating the Real Roots of an Equation 225
Exercises on Chapter VI -. . . . 229
Chapter 7. COMPLEX NUMBERS. POLYNOMIALS
1. Complex Numbers. Basic Definitions 233
2. Basic Operations on Complex Numbers : 234
three. Powers and Roots of Circuitous Numbers 237
4. Exponential Function with Complex Exponent and Its Properties . . 240
five. 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
nine. Interpolation. Lagrange'south Interpolation Formula 250
ten. On the Best Approximation of Functions by Polynomials. Chebyshev's Theory . . . . . 252
Exercises on Chapter VII 253
Chapter VIII. FUNCTIONS OF SEVERAL VARIABLES
ane. Definition of a Function of Several Variables 255
2. Geometric Representation of a Function of Two Variables ….. 25§
three. Fractional and Full Increment of a Part 259
4. Continuity of a Function of Several Variables 260
5. Partial Derivatives of a Function of Several Variables 263
vi. The Geometric Interpretation of the Fractional Derivatives of a Function of Ii Variables 264
seven. Full Increment and Total Differentials 265
8. Approximation by Total Differentials 268
9. Error Approximation by Differentials 270
ten. The Derivative of a Composite Function. The Total Derivative . . . 273
11. The Derivative of a Role Defined Implicitly 276
12. Partial Derivatives of Different Orders 279
thirteen. Level Surfaces .283
14. Directional Derivatives . 284
15. Gradient 286
16. Taylor's Formula for a Function of 2 Variables 290
17. Maximum and-Minimum of a Function of Several Variables …. 292
18. Maximum and Minimum of a Part of Several Variables Related
past Given Equations (Provisional Maxima and Minima) 300
xix. Singular Points of a Curve 305
Exercises on Chapter VIII 310
Affiliate Ix. APPLICATIONS OF DIFFERENTIAL CALCULUS TO SOLID GEOMETRY
1. The Equations of a Curve in Infinite 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
four. The Kickoff 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
ane. Antiderivative and the Indefinite Integral 342
ii. 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
seven. Rational Fractions. Fractional Rational Fractions and Their Integration 357
8. Decomposition of a Rational Fraction into Fractional Fractions …. 361
nine. Integration of Rational Fractions 365
x. Ostrogradsky'southward Method 368
11. Integrals of Irrational Functions 371
12. Integrals of the Form [R(x> 5'ax2+bx + c)dx . 372
13. Integration of Binomial Differentials 375
xiv. Integration of Certain Classes of Trigonometric Functions 378
fifteen. Integration of Certain Irrational Functions past Ways of Trigonometric Substitutions 383
16. Functions Whose Integrals Cannot Exist Expressed in Terms of Simple Functions 385
Exercises on Affiliate X 386
Affiliate XI. THE DEFINITE INTEGRAL
1. Statement of the Problem. The Lower and Upper Integral Sums . . . 396
two. The Definite Integral 398
iii. Basic Properties o? the Definite Integral . 404
4. Evaluating a Definite Integral. Newton-Leibniz Formula 407
5. Irresolute the Variable in the Definite Integral 412
6. Integration by Parts 413
7. Improper Integrals 416
viii. Approximating Definite Integrals 424
9. Chebyshev'southward 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. Calculating Areas in Rectangular Coordinates 442
ii. The Area of a Curvilinear Sector in Polar Coordinates ' . 445
3. The Arc Length of a Bend . 447
4. Computing the Volume of a Solid from the Areas of Parallel Sections (Volumes past Slicing) 453
five. 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 Affiliate XII 462
Chapter Thirteen. 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
two. Definitions \ . . , 472
3. Commencement-Guild Differential Equations (General Notions) . 473
4. Equations with Separated and Separable Variables. The Problem of the Disintegration of Radium 478
v. Homogeneous First-Club Equations 482
6. Equations Reducible to Homogeneous Equations 484
vii. Showtime-Order Linear Equations 487
8. Bernoulli's Equation 490
ix. Verbal Differential Equations 492
10. Integrating Factor 495
11. The Envelope of a Family unit of Curves 497
12. Atypical Solutions of a First-Order Differential Equation 504
13. Clairaut's Equation 505
14. Lagrange'southward Equation 507
fifteen. Orthogonal and Isogonal Trajectories 509
sixteen. Higher-Order Differential Equations (Fundamentals) 514
17. An Equation of the Form y<northward> = f(x) 516
18. Some Types of 2d-Club Differential Equations Reducible to Kickoff-Society Equations . . 518
19. Graphical Method of Integrating Second-Club Differential Equations 527
20. Homogeneous Linear Equations. Definitions and Full general Backdrop 528
21. Second-Order Homogeneous Linear Equations with Abiding Coefficients ( 535
22. Homogeneous Linear Equations of the Aith Order with Abiding Coefficients 539
23. Nonhomogeneous Second-Order Linear Equations 541
24. Nonhomogeneous Second-Order Linear Equations with Abiding Coefficients 545
25. Higher-Lodge Nonhomogeneous Linear Equations 551
26. The Differential -Equation of Mechanical Vibrations 555
27. Complimentary 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'southward Theory of Stability 576
32. Euler's Method of Estimate Solution of Kickoff-Society 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 Kickoff-Order Differential Equations 591
Exercises on Chapter Thirteen 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 past Ways of Double Integrals …. 623
five. The Double Integral in Polar Coordinates 626
6. Changing Variables in a Double Integral (General Case) 633
7. Computing the Expanse of a Surface 638
8. The Density of Distribution of Matter and the Double Integral . . . 642
9. The Moment of Inertia of the Expanse of a Plane Figure 643
x. 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 Middle of Gravity of a Solid . 660
15. Calculating Integrals Dependent on a Parameter . 662
Exercises on Affiliate Fourteen . . . 663
Chapter XV. LINE INTEGRALS AND SURFACE INTEGRALS
ane. Line Integrals 670
2. Evaluating a Line Integral 673
three. Dark-green'due south Formula 679
four. Weather for a Line Integral Beingness Independent of the Path of Integration 681
5. Surface Integrals . 687
half-dozen. Evaluating Surface Integrals 689
7. Stokes' Formula 692
8. Ostrogradsky's Formula 697
nine. The Hamiltonian Operator and Certain Applications of It 700
Exercises on Affiliate XV 703
Affiliate Xvi. Series
1. Serial. Sum of a Series 710
2. Necessary Condition for Convergence of a Series 713
3. Comparison Serial with Positive Terms 716
four. D'Alembert'southward Examination 718
5. Cauchy'due south Exam 721
6. The Integral Test for Convergence of a Series .723
7. Alternating Serial. Leibniz' Theorem 727
8. PIus-and-Minus Serial. Absolute and Contitional Convergence …. 729
nine. Functional Series 733
10. Majorised Serial 734
11. The Continuity of the Sum of a Serial 736
12. Integration and Differentiation of Series 739
xiii. Ability Serial. Interval of Convergence 742
14. Differentiation of Power Series 747
xv. Series in Powers of ■* —a 748
sixteen. Taylor'southward Serial and Maclaurin's Serial .' 750
17. Examples of Expansion of Functions in Serial 751
xviii. 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 Apply of Series (Calculating Definite Integrals) …. 758
22. Integrating Differential Equations by Ways of Series ……. 760
23. Bessel'south Equation 763
Exercises on Chapter 16 768
Chapter XVII. FOURIER Serial
i. Definition. Statement of the Problem 776
2. Expansions of Functions in Fourier Series 780
iii. A Remark on the Expansion of a Periodic Role in a Fourier Series 785
4. Fourier Series for Fifty-fifty and Odd Functions 787
v. The Fourier Serial for a Function with Catamenia two/ 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
viii. The Dirichlet Integral 798
nine. The Convergence of a Fourier Series at a Given Point . . . . . . .801
x. Certain Sufficient Conditions for the Convergence of a Fourier Serial 802
11. Applied Harmonic Assay 805
12. Fourier Integral 810
xiii. The Fourier Integral in Circuitous Form 810
Exercises on Affiliate XVII 812
Chapter 18. EQUATIONS OF MATHEMATICAL PHYSICS
ane. Basic Types of Equations of Mathematical Physics . 815
two. Derivation of the Equation of Oscillations of a String. Formulation of the Purlieus-Value Trouble. Derivation of Equations of Electrical Oscillations in Wires : 816
3. Solution of the Equation of Oscillations of a Cord by the Method
of Separation of Variables (The Fourier Method) 820
iv. The Equation for Propagation of Heat in a Rod. Conception of the Boundary-Value Problem 823
5. Heat Propagation in Space 825
six. Solution of the First Boundary-Value Problem for the Heat- Conductivity Equation by the Method of Finite Differences 829
7. Propagation of Estrus in an Unbounded Rod 831
viii. Problems That Reduce to Investigating Solutions of the Laplace Equation. Stating Purlieus-Value Bug 836
9. The Laplace Equation in Cylindrical Coordinates. Solution of the Dirichlet Trouble 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
eleven. Solution of the Dirichlet Problem past the Method of Finite Differences 847
Exercises on Affiliate Xviii 850
Chapter 19. OPERATIONAL CALCULUS AND Sure OF ITS APPLICATIONS
1. The Initial Role 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
five. The Shift Theorem r 858
half dozen. Transforms of the Functionse~at, sinh at, cosh at, e-aisinatt e~a/ cos at 858
vii. Differentiation of Transforms 860
8. The Transforms of Derivatives 861
9. Table of Transforms ~. 862
x. 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 past the Operational /Method 869
13. The Convolution Theorem 871
14. The Differential Equations of Mechanical Oscillations. The Differential Equations of Electric-Excursion Theory 873
fifteen. 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
xix. The Filibuster Theorem 879
Exercises on Chapter 19 880
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