In this post, we will see the much awaited two volume set *Differential and Integral Calculus* by N. Piskunov.

# About the book

Text book by the late professor Nikolai Piskunov DSs (Physics and Maths) is devoted to the most important divisions of higher mathematics. This edition revised and last published in two volumes

The first volume dealing with the following topics: Number, Variable, Function, Limit, Continuity of a Function, Derivative and Differential, Certain Theorems on Differentiable 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, Mechanical Applications of the Definite Integral.

The second volume dealing with the following topics: Differential Equations, Multiple Integrals, Line and Surface Integrals, Series, Fourier Series, The Equations of Mathematical Physics, Operational Calculus and Certain of its Applications, Elements of the Theory of Probability and Mathematical Statistics, Matrices.

There are numerous examples and problems in each section of the course many of them demonstrate the ties between mathematics and other senses making the book useful for self study is a textbook for higher technical schools that has gone through several editions in Russian and also has been translated into French and Spanish and Portuguese.

The books were translated from the Russian by George Yankovsky and published by Mir in 2 volume format in 1981 as Fourth reprint. We have earlier seen the one volume version.

Many thanks to Ranjan G. for donating the two volume set for scanning (I only had the one volume version) and @hawakajhonka for making them available.

Credits to original uploaders the French, Spanish and Portuguese versions.

## English Version

## Versión en Español

## Version Française

## Versão em Português

# Contents

# Volume 1

## CHAPTER I. NUMBER. VARIABLE. FUNCTION

1.1 Real numbers. Real numbers as points on a number scale 11

1.2 The absolute value of a real number 12

1.3. Variables and constants 14

1.4 The range of a variable 14

1.5 Ordered variables. Increasing and decreasing variables. Bounded Variables 16

1.6 Function 16

1.7 Ways of representing functions 18

1.8 Basic elementary functions. Elementary functions 20

1.9 Algebraic functions 24

1.10 Polar coordinate system 26

Exercises on Chapter 27

## CHAPTER 2. LIMIT. CONTINUITY OF A FUNCTION

2.1 The limit of a variable. An infinitely large variable 29

2.2 The limit of a function 31

2.3 A function that approaches infinity. Bounded functions 35

2.4 Infinitesimals and their basic properties 39

2.5 Basic theorems on limits 42

2.6 The limit of the function *sin x / x* as *x* → 0 46

2.0. The number *e* 47

2.8 Natural logarithms 51

2.9 Continuity of functions 53

2.10 Certain properties of continuous functions 57

2.11 Comparing infinitesimals 59

Exercises on Chapter 2 61

## CHAPTER 3. DERIVATIVE AND DIFFERENTIAL

3.1 Velocity of motion 65

3.2 The definition of a derivative 67

3.3 Geometric meaning of the derivative 69

3.4 Differentiability of functions 70

3.5 The derivative of the function *y=x^{n}*, *n* a positive integer 74

3.6 Derivatives of the functions* y = sin x, y = cos x* 75

3.7 Derivatives of: a constant, the product of a constant by a function, a sum, a product, and a quotient 75

3.8 The derivative of a logarithmic function 80

3.9 The derivative of a composite function 81

3.10 Derivatives of the functions *y = tan x, y = cot x, y = ln |x|* 83

3.11 An Implicit function and its differentiation 85

3.12 Derivatives of a power function for an arbitrary real exponent, of general exponential function, and of a composite exponential function 87

3.13 An inverse function and its differentiation 89

3.14 Inverse trigonometric functions and their differentiation 92

3.15 Basic differentiation formulas 96

3.16 Parametric representation of a function 98

3.17 The equations of some curves in parametric form 99

3.18 The derivative of a function represented parametrically 102

3.19 Hyperbolic functions 104

3.20 The differential. 107

3.21 The geometric meaning of the differential 111

3.22 Derivatives of different orders 112

3.23 Differentials of different orders 114

3.24 Derivatives (of various orders) of implicit functions and of functions represented parametrically 116

3.25 The mechanical meaning of the second derivative 118

3.26 The equations of a tangent and of a normal. The lengths of a subtangent and a subnormal 119

3.27 The geometric meaning of the derivative of the radius vector with respect to the polar angle 122

Exercises on Chapter 3

CHAPTER 4. SOME THEOREMS ON DIFFERENTIABLE FUNCTIONS

4.1 A theorem on the roots of a derivative (Rolle’s theorem) 133

4.2 The mean-value theorem (Lagrange’s theorem) 135

4.3 The generalized mean-value theorem (Cauchy’s theorem) 136

4.4 The limit of a ratio of two infinitesimals (evaluating indeterminate forms of the type 0/0 137

4.5 The limit of a ratio of two infinitely large quantities(evaluating indeterminate forms of the type ∞/∞) 140

4.6 Taylor’s formula 145

4.7 Expansion of the functions *e^{x}*, *sin x*, and *cos x* in a Taylor series 149

Exercises on Chapter 4 152

CHAPTER 5. INVESTIGATING THE BEHAVIOUR OF FUNCTIONS

5.1Statement of the problem 155

5.2 Increase and decrease of a function 156

5.3 Maxima and minima of functions 157

5.4 Testing a differentiable function for maximum and minimum with a first derivative 164

5.5 Testing a function for maximum and minimum with a second derivative 166

5.6 Maximum and minimum of a function on an interval 170

5.7 Applying the theory of maxima and minima of functions to the solution of problems 171

5.8 Testing a function for maximum and minimum by means of Taylor’s formula 173

5.9 Convexity and concavity of a curve. Points of inflection 175

5.10 Asymptotes 182

5.11 General plan for investigating functions and constructing graphs 186

5.12 Investigating curves represented parametrically 190

Exercises on Chapter 5 194

CHAPTER 6. THE CURVATURE OF A CURVE

6.1 Arc length and its derivative 200

6.2 Curvature 202

6.3 Calculation of curvature 204

6.4 Calculating the curvature of a curve represented parametrically 207

6.5 Calculating the curvature of a curve given by an equation in polar Coordinates 207

6.6 The radius and circle of curvature. The centre of curvature. Evolute and involute 208

6.7 The properties of an evolute 213

6.8 Approximating the real roots of an equation 216

Exercises on Chapter 6 221

## CHAPTER 7. COMPLEX NUMBERS. POLYNOMIALS

7.1 Complex numbers. Basic definitions 224

7.2 Basic operations on complex numbers 226

7.3 Powers and roots of complex numbers 229

7.4 Exponential function with complex exponent and its properties 231

7.5 Euler’s formula. The exponential form of a complex number 234

7.6 Factoring a polynomial 235

7.7 The multiple roots of a polynomial 238

7.8 Factoring a polynomial in the case of complex roots 240

7.9 Interpolation. Lagrange’s interpolation formula 241

7.10 Newton’s interpolation formula 243

7.11 Numerical differentiation 245

7.12 On the best approximation of functions by polynomials. Chebyshev’s theory 246

Exercises on Chapter 7 247

CHAPTER 8. FUNCTIONS OF SEVERAL VARIABLES

8.1 Definition of a function of several variables 249

8.2 Geometric representation of a function of two variables 252

8.3 Partial and total increment of a function 253

8.4 Continuity of a function of several variables 254

8.5 Partial derivatives of a function of several variables 257

8.6 A geometric interpretation of the partial derivatives of a function of two variables 259

8.7 Total increment and total differential 260

8.8 Approximation by total differentials 263

8.9 Use of a differential to estimate errors in calculations 264

8.10 The derivative of a composite function. The total derivative. The total differential of a composite function 267

8.11 The derivative of a function defined implicitly 270

8.12 Partial derivatives of higher orders 273

8.13 Level surfaces 277

8.14 Directional derivative 278

8.15 Gradient 281

8.16 Taylor’s formula for a function of two variables 284

8.17 Maximum and minimum of a function of several variables 286

8.18 Maximum and minimum of a function of several variables related by given equations (conditional maxima and minima) 293

8.19 Obtaining a function on the basis of experimental data by the method of least squares 298

8.20 Singular points of a curve 302

Exercises on Chapter 8 307

CHAPTER 9. APPLICATIONS OF DIFFERENTIAL CALCULUS TO SOLID GEOMETRY

9.1 The equations of a curve in space 311

9.2 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 314

9.3 Rules for differentiating vectors (vector functions) 320

9.4 The first and second derivatives of a vector with respect to arc length. The curvature of a curve. The principal normal. The velocity and acceleration of a point in curvilinear motion 322

9.5 Osculating plane. Binormal. Torsion 330

9.6 The tangent plane and the normal to a surface 335

Exercises on Chapter 9 338

CHAPTER 10. THE INDEFINITE INTEGRAL

10.1 Antiderivative and the indefinite integral 341

10.2 Table of integrals 343

10.3 Some properties of the indefinite integral 345

10.4 Integration by substitution (change of variable) 347

10.5 Integrals of some functions containing a quadratic trinomial 350

10.6 Integration by parts 352

10.7 Rational fractions. Partial rational fractions and their integration 356

10.8 Decomposition of a rational fraction into partial fractions 359

10.9 Integration of rational fractions 363

10.10 Integrals of irrational functions 366

10.11 Integrals of the form ∫R(x,√(ax^2+bx+c)) dx 367

10.12 Integration of certain classes of trigonometric functions 370

10.13 Integration of certain irrational functions by means of trigonometric substitutions 375

10.14 On functions whose integrals cannot be expressed in terms of elementary functions 377

Exercises on Chapter 10 378

CHAPTER 11. THE DEFINITE INTEGRAL

11.1 Statement of the problem. Lower and upper sums 387

11.2 The definite integral. Proof of the existence of a definite integral 389

11.3 Basic properties of the definite integral 399

11.4 Evaluating a definite integral. The Newton-Leibniz formula 402

11.5 Change of variable in the definite integral 407

11.6 Integration by parts 408

11.7 Improper integrals 411

11.8 Approximating definite integrals 419

11.9 Chebyshev’s formula 424

11.10 Integrals dependent on a parameter. The gamma function 429

11.11 Integration of a complex function of a real variable 433

Exercises on Chapter 11 433

CHAPTER 12. GEOMETRIC AND MECHANICAL APPLICATIONS OF THE DEFINITE INTEGRAL

12.1 Computing areas in rectangular coordinates 437

12.2 The area of a curvilinear sector in polar coordinates 440

12.3 The arc length of a curve 441

12.4 Computing the volume of a solid from the areas of parallel sections (volumes by slicing) 447

12.5 The volume of a solid of revolution 449

12.6 The surface of a solid of revolution 450

12.7 Computing work by the definite integral 452

12.8 Coordinates of the centre of gravity 453

12.9 Computing the moment of inertia of a line, a circle, and a cylinder by means of a definite integral 456

Exercises on Chapter 12 458

Index 465

# Volume 2

## CHAPTER 1 DIFFERENTIAL EQUATIONS

1.1 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 11

1.2 Definitions 14

1.3 First-order differential equations (general notions) 15

1.4 Equations with separated and separable variables. The problem of disintegration of radium 20

1.5 Homogeneous first-order equations 24

1.6 Equations reducible to homogeneous equations 26

1.7 First-order linear equations 29

1.8 Bernoulli’s equation 32

1.9 Exact differential equations 34

1.10 Integrating factor 37

1.11 The envelope of a family of curves 39

1.12 Singular solutions of a first-order differential equation 45

1.13 Clairaut’s equation 46

1.14 Lagrange’s equation 48

1.15 Orthogonal, and isogonal trajectories 50

1.16 Higher-order differential equations (fundamentals) 55

1.17 An equation of the form y^{(n)} = f ( x ) 56

1.18 Some types of second-order differential equations reducible to first-order equations. Escape-velocity problem 59

1.19 Graphical method of integrating second-order differential equations 66

1.20 Homogeneous linear equations. Definitions and general properties 68

1.21 Second-order homogeneous linear equations with constant coefficients 75

1.22 Homogeneous linear equations of the nth order with constant coefficients 80

1.23 Nonhomogeneous second-order linear equations 82

1.24 Nonhomogeneous second-order linear equations with constant coefficients 86

1.25 Higher-order nonhomogeneous linear equations 93

1.26 The differential equation of mechanical vibrations 97

1.27 Free oscillations 98

1.28 Forced oscillations 102

1.29 Systems of ordinary differential equations 106

1.30 Systems of linear differential equations with constant coefficients 111

1.31 On Lyapunov’s theory of stability 117

1.32 Euler’s method of approximate solution of first-order differential equations 133

1.33 A difference method for approximate solution of differential equations based on Taylor’s formula. Adams method 142

1.34 An approximate method for integrating systems of first-order differential equations 146

Exercises on Chapter 1 146

## CHAPTER 2 MULTIPLE INTEGRALS

2.1 Double integrais 158

2.2 Calculating double integrais 161

2.3 Calculating double integrals (continued) 166

2.4 Calculating areas and volumes by means of double integrals 172

2.5 The double integral in polar coordinates 175

2.6 Change of variables in a double integral(general case) 182

2.7 Computing the area of a surface 187

2.8 The density distribution of matter and the double integral 190

2.9 The moment of inertia of the area of a plane figure 191

2.10 The coordinates of the centre of gravity of the area of a plane figure 196

2.11 Triple integrais 197

2.12 Evaluating a triple integral 198

2.13 Change of variables in a triple integral 204

2.14 The moment of inertia and the coordinates of the centre of gravity of a solid 207

2.15 Computing integrais dependent on a parameter 209

Exercises on Chapter 2 211

CHAPTER 3 LINE INTEGRALS AND SURFACE INTEGRALS

3.1 Line integrals 216

3.2 Evaluating a line integral 219

3.3 Green’s formula 225

3.4 Conditions for a line integral to be independent of the path of integration 227

3.5 Surface integrals 232

3.6 Evaluating surface integrals 234

3.7 Stokes* formula 236

3.8 Ostrogradsky’s formula 241

3.9 The Hamiltonian operator and some applications 244

Exercises on Chapter 3

## CHAPTER 4 SERIES

4.1 Series. Sum of a series 253

4.2 Necessary condition for convergence of a series 256

4.3 Comparing series with positive terms 258

4.4 D’Alembert’s test 260

4.5 Cauchy’s test 264

4.6 The integral test for convergence of a series 266

4.7 Alternating series. Leibniz theorem 269

4.8 Plus-and-minus series. Absolute and conditional convergence 271

4.9 Functional series 274

4.10 Decimated series 275

4.11 The continuity of the sum of a series 277

4.12 Integration and differentiation of series 280

4.13 Power series. Interval of convergence 283

4.14 Differentiation of power series 288

4.15 Series in powers of x – a 289

4.16 Taylor’s series and Maclaurin’s series 290

4.17 Series expansion of functions 292

4.18 Euler’s formula 294

4.19 The binomial series 295

4.20 Expansion of the function ln( 1 + x ) in a power series. Computing logarithms 297

4.21 Series evaluation of definite integrals 299

4.22 Integrating differential equations by means of series 301

4.23 Bessel’s equation 303

4.24 Series with complex terms 308

4.25 Power series in a complex variable 309

4.26 The solution of first-order differential equations by the method of successive approximations (method of iteration) 312

4.27 Proof of the existence of a solution of a differential equation. Error estimation in approximate solutions 313

4.28 The uniqueness theorem of the solution of a differential equation 318

Exercises on Chapter 4 319

## CHAPTER 5 FOURIER SERIES

5.1 Definition. Statement of the problem 327

5.2 Expansions of functions in Fourier series 331

5.3 A remark on the expansion of a periodic function in a Fourier series 336

5.4 Fourier series for even and odd functions 338

5.5 The Fourier series for a function with period 339

5.6 On the expansion of a nonperiodic function in aFourier series 341

5.7 Mean approximation of a given function by a trigonometric polynomial 343

5.8 The Dirichlet integral 348

5.9 The convergence of a Fourier series at a given point 351

5.10 Certain sufficient conditions for the convergence of a Fourier series 352

5.11 Practical harmonic analysis 355 5.12 The Fourier series in complex form 356

5.13 Fourier integral 358

5.14 The Fourier integral in complex form 362

5.15 Fourier series expansion with respect to an orthogonal System of functions 364

5.16 The concept of a linear function space. Expansion of functions in Fourier series compared with decomposition of vectors 367

Exercises on Chapter 5 371

## CHAPTER 6 EQUATIONS OF MATHEMATICAL PHYSICS

6.1 Basic types of equations of mathematical physics 374

6.2 Deriving the equation of the vibrating string. Formulating the boundary-value problem. Deriving equations of electric oscillations in

wires 375

6.3 Solution of the equation of the vibrating string by the method of separation of variables (the Fourier method ) 378

6.4 The equation of heat conduction in a rod. Formulation of the boundary-value problem 382

6.5 Heat transfer in space 384

6.6 Solution of the first boundary-value problem for the heat-conduction equation by the method of finite differences 387

6.7 Heat transfer in an unbounded rod 389

6.8 Problems that reduce to investigating solutions of the Laplace equation. Stating boundary-value problems 394

6.9 The Laplace equation in cylindrical coordinates. Solution of the Dirichlet problem for an annulus with constant values of the desired function on the inner and outer circumferences 399

6.10 The solution of Dirichlet’s problem for a circle 401

6.11 Solution of the Dirichlet problem by the method of finite differences 405

Exercises on Chapter 6 407

CHAPTER 7 OPERATIONAL CALCULAIS AND CERTAIN OF ITS APPLICATIONS

7.1 The original function and its transform 411

7.2 Transforms of the functions 𝜎_{0}(t). sin t, cos t 413

7.3 The transform of a function with changed scale of the independent variable. Transforms of the functions sin at, cos at 414

7.4 The linearity property of a transform 415

7.5 The shift theorem 416

7.6 Transforms of the functions e^{-𝛼t}, sinh 𝛼t, cosh 𝛼t, e^{-𝛼t} sin 𝛼t, e^{-𝛼t} cos 𝛼t 416

7.7 Differentiation of transforms 417

7.8 The transforms of derivatives 419

7.9 Table of transforms 420

7.10 An auxiliary equation for a given differential equation 422

7.11 Decomposition theorem 426

7.12 Examples of solutions of differential equations and Systems of differential equations by the operational method 428

7.13 The convolution theorem 429

7.14 The differential equations of mechanical vibrations. The differential equations of electric-circuit theory 432

7.15 Solution of the differential equation of oscillations 433

7.16 Investigating free oscillations 435

7.17 Investigating mechanical and electrical oscillations in the case of a periodic external force 435

7.18 Solving the oscillation equation in the case of resonance 437

7.19 The delay theorem 439

7.20 The delta function and its transform 440

Exercises on Chapter 7 443

CHAPTER 8 ELEMENTS OF THE THEORY OF PROBABILITY AND MATHEMATICAL STATISTICS

8.1 Random event. Relative frequency of a random event. The probability of an event. The subject of probability theory 445

8.2 The classical definition of probability and the calculation of probabilites 447

8.3 The addition of probabilites. Complementary random events 449

8.4 Multiplication of probabilites of independent e v e n t s 452

8.5 Dependent events. Conditional probability. Total probability 454

8.6 Probability of causes. Bayes’s formula 457

8.7 A discrète random variable. The distribution law of a discrète random variable 460

8.8 Relative frequency and the probability of relative frequency in repeated trials 462

8.9 The mathematical expectation of a discrète random variable 466

8.10 Variance. Root-mean-square (standard) deviation. Moments 471

8.11 Functions of random variables 474

8.12 Continuous random variable. Probability density function of a continuous random variable. The probability of the random variable falling in a specified interval 475

8.13 The distribution function. Law of uniform distribution 479

8.14 Numerical characteristics of a continuous random variable 482

8.15 Normal distribution. The expectation of a normal distribution 485

8.16 Variance and standard deviation of a normally distributed random variable 487

817 The probability of a value of the random variable falling in a given interval. The Laplace function. Normal distribution function 488

8.18 Probable error 493

8.19 Expressing the normal distribution in terms of the probable error. The reduced Laplace function 494

8.20 The three-sigma rule. Error distribution 496

8.21 Mean arithmetic error 497

8.22 Modulus of precision. Relationships between the characteristics of the distribution of errors 498

8.23 Two-dimensional random variables 499

8.24 Normal distribution in the plane 502

8.25 The probability of a two-dimensional random variable falling in a rectangle with sides parallel to the principal axes of dispersion

under the normal distribution law 504

8.26 The probability of a two-dimensional random variable falling in the ellipse of dispersion 506

8.27 Problems of mathematical statistics. Statistical data 507

8.28 Statistical series. Histogram 508

8.29 Determining a suitable value of a measured quantity 511

8.30 Determining the parameters of a distribution law. Lyapunov’s theorem. Laplace’s theorem 512

Exercises on Chapter 8 516

## CHAPTER 9 MATRICES

9.1 Linear transformations. Matrix notation 519

9.2 General definitions involving matrices 522

9.3 Inverse transformation 524

9.4 Operations on matrices. Addition of matrices 526

9.5 Transforming a vector into another vector by means of a matrix 529

9.6 Inverse matrix 531

9.7 Matrix inversion 532

9.8 Matrix notation for Systems of linear equations and solutions of systems of linear equations 534

9.9 Solving Systems of linear equations by the matrix method 535

9.10 Orthogonal mappings. Orthogonal matrices 537

9.11 The eigenvector of a linear transformation 540

9.12 The matrix of a linear transformation under which the base vectors

are eigenvectors 543

9.13 Transforming the matrix of a linear transformation when changing

the basis 544

9.14 Quadratic forms and their transformation 547

9.15 The rank of a matrix. The existence of solutions of a system of linear equations 549

9.16 Differentiation and integration of matrices 550

9.17 Matrix notation for Systems of differential equations and solutions

of Systems of differential equations with constant coefficients 552

9.18 Matrix notation for a linear equation of order n 557

9.19 Solving a System of linear differential equations with variable coefficients by the method of successive approximations using matrix

notation 558

Exercises on Chapter 9 563

Appendix 565

Index 567

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