In this post, we will see the book Introductory Mathematics for Engineers: Lectures in Higher Mathematics by A. D. Myškis.About The BookThe present book is based on lectures given by the author over a number of years to students of various engineering and physics. The book includes some optional can be skipped for the first reading. The corresponding Items m the table of contents are marked by an asterisk....The book is composed in such a way that it is possible to use it both for studying in a college under the guidance of a teacher and for self-education. The subject matter of the book is divided into small sections so that the reader could study the material in suitable order and to any extent depending on the profession and the needs of the reader. It is also intended that the book can be used by students taking a correspondence course and by the readers who have some prerequisites in higher mathematics and want to perfect their knowledge by reading some chapters of the book....The book can be of use to readers of various professions dealing with applications of mathematics in their work. Modern applied mathematics of many important special divisions which are not included m this book. The author intends to write another book devoted to some supplementary topics such as the theory of functions of a complex argument, variational calculus, mathematical physics, some special questions of the theory of ordinary differential equations and so on. The book has interesting ways to treat affine mappings (pages 344-345) and non-linear mappings (pages 358-359).The book was translated from the Russian by V. M. Volosov and was first published by Mir in 1972.Note: quite a few pages are missing from the scan:56-57 70-71  210-211  240-241  312-313  315  320-321  337-338 338-339-340  418-419  464-465  759-760  764-765Credits to the original scanner. The original scan was not clean or bookmarked. We cleaned, OCRed and bookmarked the original scan.ContentsFront CoverTitle PagePreface 5ContentsIntroduction 191. The Subject of Mathematics 192. The Importance of Mathematics and Mathematical Education 203. Abstractness 204. Characteristic Features of Higher Mathematics 225. Mathematics in the Soviet Union 23CHAPTER I. VARIABLES AND FUNCTIONS 25§ 1. Quantities 251. Concept of a Quantity 252. Dimensions of Quantities 253. Constants and Variables 264. Number Scale. Slide Rule 275. Characteristics of Variables 29§ 2. Approximate Values of Quantities 326. The Notion of an Approximate Value 327. Errors 328. Writing Approximate Numbers 339. Addition and Subtraction of Approximate Numbers 3410. Multiplication and Division of Approximate Numbers Remarks 36§ 3. Functions and Graphs 3911. Functional Relation 3912. Notation 4013. Methods of Representing Functions 4214. Graphs of Functions 4515. The Domain of Definition of a Function 4716. Characteristics of Behaviour of Functions 4817. Algebraic Classification of Functions 5118. Elementary Functions 5319. Transforming Graphs 5420. Implicit Functions 5621. Inverse Functions 58§ 4. Review of Basic Functions 6022. Linear Function 6023. Quadratic Function 6224. Power Function 6325. Linear-Fractional Function 6626. Logarithmic Function 6827. Exponential Function 6928. Hyperbolic Functions 7029. Trigonometric Functions 7230. Empirical Formulas 75CHAPTER II. PLANE ANALYTIC GEOMETRY 78§ 1. Plane Coordinates 781. Cartesian Coordinates 782. Some Simple Problems Concerning Cartesian Coordinates 793. Polar Coordinates 81§ 2. Curves in Plane 824. Equation of a Curve in Cartesian Coordinates 825. Equation of a Curve in Polar Coordinates 846. Parametric Representation of Curves and Functions 877. Algebraic Curves 908. Singular Cases 92§ 3. First-Order and Second-Order Algebraic Curves 949. Curves of the First Order 9410. Ellipse 9611. Hyperbola 9912. Relationship Between Ellipse, Hyperbola and Parabola 10213. General Equation of a Curve of the Second Order 105CHAPTER III. LIMIT. CONTINUITY 109§ 1. Infinitesimal and Infinitely Large Variables 1091. Infinitesimal Variables 1092. Properties of Infinitesimals 1113. Infinitely Large Variables 112§ 2. Limits 1134. Definition 1135. Properties of Limits 1156. Sum of a Numerical Series 117§ 3. Comparison of Variables 1217. Comparison of Infinitesimals 1218. Properties of Equivalent Infinitesimals 1229. Important Examples 12210. Orders of Smallness 12411. Comparison of Infinitely Large Variables 125§ 4. Continuous and Discontinuous Functions 12512. Definition of a Continuous Function 12513. Points of Discontinuity 12614. Properties of Continuous Functions 12915. Some Applications 131CHAPTER IV. DERIVATIVES, DIFFERENTIALS, INVESTIGATION OF THE BEHAVIOUR OF FUNCTIONS 134§ 1. Derivative 1341. Some Problems Leading to the Concept of a Derivative 1342. Definition of Derivative 1363. Geometrical Meaning of Derivative 1374. Basic Properties of Derivatives 1395. Derivatives of Basic Elementary Functions 1426. Determining Tangent in Polar Coordinates 146§ 2. Differential 1487. Physical Examples 1488. Definition of Differential and Its Connection with Increment 1499. Properties of Differential 15210. Application of Differentials to Approximate Calculations 153§ 3. Derivatives and Differentials of Higher Orders 15511. Derivatives of Higher Orders 15512. Higher-Order Differentials 156§ 4. L'Hospital's Rule 15813. Indeterminate Forms of the Type $\dfrac{0}{0}$ 15814. Indeterminate Forms of tl1e Type $\dfrac{\infty}{\infty}$ 160§ 5. Taylor's Formula and Series 16115. Taylor's Formula 16116. Taylor's Series 163§ 6. Intervals of Monotonicity. Exrtremum 16517. Sign of Derivative 16518. Points of Extremum 16619. The Greatest and the Least Values of a Function 168§ 7. Constructing Graphs of Functions 17320. Intervals of Convexity of a Graph and Points of Inflection 17321. Asymptotes of a Graph 17422. General Scheme for Investigating a Function and Constructing Its Graph 175CHAPTER V. APPROXIMATING ROOTS OF EQUATIONS. INTERPOLATION 179§ 1. Approximating Roots of Equations 1791. Introduction 1792. Cut-and-Try Method. Method of Chords. Method of Tangents 1813. Iterative Method 1854. Formula of Finite Increments 1875*. Small Parameter Method 189§ 2. Interpolation 1916. Lagrange's Interpolation Formula 1917. Finite Differences and Their Connection with Derivatives 1928. Newton's Interpolation Formulas 1969. Numerical Differentiation 198CHAPTER VI. DETERMINANTS AND SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS 200§ 1. Determinants 2001. Definition 2002. Properties 2013. Expanding a Determinant in Minors of Its Row or Column 203§ 2. Systems of Linear Algebraic Equations 2064. Basic Case 2065. Numerical Solution 2086. Singular Case 209CHAPTER VII. VECTORS 212§ 1. Linear Operations on Vectors 2121. Scalar and Vector Quantities 2122. Addition of Vectors 2133. Zero Vector and Subtraction of Vectors 2154. Multiplying a Vector by a Scalar 2155. Linear Combination of Vectors 216§ 2. Scalar Product of Vectors 2196. Projection of Vector on Axis 2197. Scalar Product 2208. Properties of Scalar Product 221§ 3. Cartesian Coordinates in Space 2229. Cartesian Coordinates in Space 22210. Some Simple Problems Concerning Cartesian Coordinates 223§ 4. Vector Product of Vectors 22711. Orientation of Surface and Vector of an Area 22712. Vector Product 22813. Properties of Vector Products 23014*. Pseudovectors 233§ 5. Products of Three Vectors 23515. Triple Scalar Product 23516. Triple Vector Product 236§ 6. Linear Spaces 23717. Concept of Linear Space 23718. Examples 23919. Dimension of Linear Space 24120. Concept of Euclidean Space 24421. Orthogonality 245§ 7. Vector Functions of Scalar Argument. Curvature 24822. Vector Variables 24823. Vector Functions of Scalar Argument 24824. Some Notions Related to the Second Derivative 25125. Osculating Circle 25226. Evolute and Evolvent 255CHAPTER VIII. COMPLEX NUMBERS AND FUNCTIONS 259§ 1. Complex Numbers 2591. Complex Plane 2592. Algebraic Operations on Complex Numbers 2613. Conjugate Complex Numbers 2634. Euler's Formula 2645. Logarithms of Complex Numbers 266§ 2. Complex Functions of a Real Argument 2676. Definition and Properties 2677*. Applications to Describing Oscillations 269§ 3. The Concept of a Function of a Complex Variable 2718. Factorization of a Polynomial 2719*. Numerical Methods of Solving Algebraic Equations 27310. Decomposition of a Rational Fraction into Partial Rational Fractions 27711*. Some General Remarks on Functions of a Complex Variable 280CHAPTER IX. FUNCTIONS OF SEVERAL VARIABLES 283§ 1. Functions of Two Variables 2831. Methods of Representing 2832. Domain of Definition 2863. Linear Function 2874. Continuity and Discontinuity 2885. Implicit Functions 291§ 2. Functions of Arbitrary Number of Variables 2916. Methods of Representing 2917. Functions of Three Arguments 2928. General Case 2929. Concept of Field 293§ 3. Partial Derivatives and Differentials of the First Order 29410. Basic Definitions 29411. Total Differential 29612. Derivative of Composite Function 29813. Derivative of Implicit Function 300§ 4. Partial Derivatives and Differentials of Higher Orders 30314. Definitions 30315. Equality of Mixed Derivatives 30416. Total Differentials of Higher Order 305CHAPTER X. SOLID ANALYTIC GEOMETRY 307§ 1. Space Coordinates 3071. Coordinate Systems in Space 3072*. Degrees of Freedom 3094. Cylinders, Cones and Surfaces of Revolution 3145. Curves In Space 3166. Parametric Representation of Surfaces in Space. Parametric Representation of Functions of Several Variables 317§ 3. Algebraic Surfaces of the First and of the Second Orders 3197. Algebraic Surfaces of the First Order 3198. Ellipsoids 3229. Hyperboloids 32410. Paraboloids 32611. General Review of the Algebraic surfaces of the second order 327CHAPTER XI. MATRICES AND THEIR APPLICATIONS 329§ 1. Matrices 3291. Definitions 3292. Operations on Matrices 3313. Inverse Matrix 3334. Eigenvectors and Eigenvalues of a Matrix 3355. The Rank of a Matrix 3377. Transformation of the Matrix of a Linear Mapping When the Basis Is Changed 3478. The Matrix of a Mapping Relative to the Basis Consisting of Its Eigenvectors 3509. Transforming Cartesian Basis 35210. Symmetric Matrices 353§ 3. Quadratic Forms 35511. Quadratic Forms 35512. Simplification of Equations of Second-Order Curves and Surfaces 357§ 4. Non-Linear Mappings 35813*. General Notions 35814*. Non Linear Mapping in the Small 36015*. Functional Relation Between Functions 362CHAPTER XII. APPLICATIONS OF PARTIAL DERIVATIVES 365§ 1. Scalar Field 3651. Directional Derivative. Gradient 3652. Level Surfaces 3683. Implicit Functions of Two Independent Variables 3704. Plane Fields 3715. Envelope of One-Parameter Family of Curves 372§ 2. Extremum of a Function of Several Variables 3746. Taylor's Formula for a Function of Several Variables 3747. Extremum 3758. The Method of Least Squares 3809*. Curvature of Surfaces 38110. Conditional Extremum 38411. Extremum with Unilateral Constraints 38812*. Numerical Solution of Systems of Equations 390CHAPTER XIII. INDEFINITE INTEGRAL 393§ 1. Elementary Methods of Integration 3931. Basic Definitions 3932. The Simplest Integrals 3943. The Simplest Properties of an Indefinite Integral 3974. Integration by Parts 3995. Integration by Change of Variable (by Substitution) 402§ 2. Standard Methods of Integration 4046. Integration of Rational Functions 4057. Integration of Irrational Functions Involving Linear and Linear-Fractional Expressions 4078. Integration of Irrational Expressions Containing Quadratic Trinomials 4089. Integrals of Binomial Differentials 411lO. Integration of Functions Rationally Involving Trigonometric Functions 41211. General Remarks 415CHAPTER XIV. DEFINITE INTEGRAL 417§ 1. Definition and Basic Properties 4171. Examples Lending to the Concept of Definite Integral 4173. Relationship Between Definite Integral and Indefinite Integral 4264. Basic Properties of Definite Integral 4335. Integrating Inequalities 436§ 2. Applications of Definite Integral 4366. Two Schemes of Application 4367. Differential Equations with Variables Separable 4378. Computing Areas of Plane Geometric Figures 4439. The Arc Length of a Curve 44510. Computing Volumes of Solids 44711. Computing Area of Surface of Revolution 448§ 3. Numerical Integration 44812. General Remarks 44813. Formulas of Numerical Integration 450§ 4. Improper Integrals 45414. Integrals with Infinite Limits of Integration 45515. Basic Properties of Integrals with Infinite Limits 46416. Other Types of Improper Integral 46817*. Gamma Function 46818*. Beta Function 47119*. Principal Value of Divergent Integral 473§ 5. Integrals Dependent on Parameters 47420*. Proper Integrals 47421*· Improper Integrals 476§ 6. Line Integrals of Integration 47822. Line Integrals of the First Type 48223. Line Integrals of the Second Type 48424. Conditions for a Line Integral of the Second Type to Be Independent of the Path of Integration 488§ 7. The Concept of Generalized Function 48825*. Delta Function 48826*. Application to Constructing Influence Function 49227*. Other Generalized Functions 495CHAPTER XV. DIFFERENTIAL EQUATIONS 497§ 1. General Notions 4971. Examples 4972. Basic Definitions 498§ 2. First-Order Differential Equations 5003. Geometric Meaning 5004. Integrable Types of Equations 5035*. Equation for Exponential Function 5066. Integrating Exact Differential Equations 5097. Singular Points and Singular Solutions 5128. Equations Not Solved for the Derivative 5169. Method of Integration by Means of Differentiation 517§ 3. Higher-Order Equations and Systems of Differential Equations 51910. Higher-Order Differential Equations 51911*. Connection Between Higher-Order Equations and Systems of First-Order Equations 52112*. Geometric Interpretation of System of First-Order Equations 52213*. First Integrals 526§ 4. Linear Equations of General Form 52814. Homogeneous Linear Equations 52815. Non-Homogeneous Equations 53016*. Boundary-Value Problems 535§ 5. Linear Equations with Constant Coefficients 54117. Homogeneous Equations 54118. Non-Homogeneous Equations with Right-Hand Sides of Special Form 54519. Euler's Equations 54820*. Operators and the Operator Method of Solving Differential Equations 549§ 6. Systems of Linear Equations 55321. Systems of Linear Equations 55322*. Applications to Testing Lyapunov Stability of Equilibrium State 558§ 7. Approximate and Numerical Methods of Solving Differential Equations 56223. Iterative Method 56224*. Application of Taylor's Series 56425. Application of Power Series with Undetermined coefficients 56526*. Bessel's Functions 56627*. Small Parameter Method 56928*. General Remarks on Dependence of Solutions on Parameters 57229*. Methods of Minimizing Discrepancy 57530*. Simplification Method 57631. Euler's Method 57832. Runge-Kutta Method 58033. Adams Method 58234. Milne's Method 583CHAPTER XVI. Multiple Integrals 585§ 1. Definition and Basic Properties of Multiple Integrals 5851. Some Examples Leading to the Notion of a Multiple Integral 5852. Definition of a Multiple Integral 5863. Basic Properties of Multiple Integrals 5874. Methods of Applying Multiple Integrals 5895. Geometric Meaning of an Integral Over a Plane Region 591§ 2. Two Types of Physical Quantities 5926*. Basic Example. Mass and Its Density 5927*. Quantities Distributed in Space 594§ 3. Computing Multiple Integrals in Cartesian Coordinates 5968. Integral Over Rectangle 5969. Integral Over an Arbitrary Plane Region 59910. Integral Over an Arbitrary Surface 60211. Integral Over a Three-Dimensional Region 604§ 4. Change of Variables in Multiple Integrals 60512. Passing to Polar Coordinates in Plane 60513. Passing to Cylindrical and Spherical Coordinates 60614*. Curvilinear Coordinates in Plane 60815*. Curvilinear Coordinates in Space 61116*. Coordinates on a Surface 612§ 5. Other Types of Multiple Integrals 61517*. Improper Integrals 61518*. Integrals Dependent on a Parameter 61719*. Integrals with Respect to Measure. Generalized Functions 62020*. Multiple Integrals of Higher Order 622§ 6. Vector Field 62621*. Vector Lines 62622*. The Flux. of a Vector Through a Surface 62723*. Divergence 62924*. Expressing Divergence in Cartesian Coordinates 63225. Line Integral and Circulation 63426*. Rotation 63427. Green's Formula. Stokes' Formula 63828*. Expressing Differential Operations on Vector Fields in a Curvilinear Orthogonal Coordinate System 64129*. General Formula for Transforming Integrals 642CHAPTER XVII. SERIES 645§ 1. Number Series 6451. Positive Series 6452. Series with Terms of Arbitrary Signs 6503. Operations on Series 6524*. Speed of Convergence of a Series 6545. Series with Complex, Vector and Matrix Terms 6586. Multiple Series 659§ 2. Functional Series 6617. Deviation of Functions 6618. Convergence of a Functional Series 6629. Properties of Functional Series 664§ 3. Power Series 66610. Interval of Convergence 66611. Properties of Power Series 66712. Algebraic Operations on Power Series 67113. Power Series as a Taylor Series 67514. Power Series with Complex Terms 67615*. Bernoullian Numbers 67716*. Applying Series to Solving Difference Equations 67817*. Multiple Power Series 68018*. Functions of Matrices 68119*. Asymptotic Expansions 685§ 4. Trigonometric Series 68620. Orthogonality 68621. Series in Orthogonal Functions 68922. Fourier Series 69023. Expanding a Periodic Function 69524*. Example. Bessel's Functions as Fourier Coefficients 69725. Speed of Convergence of a Fourier Series 69826. Fourier Series in Complex Form 70227*. Parseval Relation 70428*. Hilbert Space 70629*. Orthogonality with Weight Function 70830*. Multiple Fourier Series 71031*. Application to the Equation of Oscillations of a String 711§ 5. Fourier Transformation 71332*. Fourier Transform 71333*. Properties of Fourier Transforms 71734*. Application to Oscillations of Infinite String 719CHAPTER XVIII. ELEMENTS OF THE THEORY OF PROBABILITY 721§ 1. Random Events and Their Probabilities 7211. Random Events 7212. Probability 7223. Basic Properties of Probabilities 7254. Theorem of Multiplication of Probabilities 7275. Theorem of Total Probability 7296*. Formulas for the Probability of HyPotheses 7307. Disregarding Low-Probability Events 731§ 2. Random Variables 7328. Definitions 7329. Examples of Discrete Random Variables 73410. Examples of Continuous Random Variables 73611. Joint Distribution of Several Random Variables 73712. Functions of Random Variables 739§ 3. Numerical Characteristics of Random Variables 74113. The Mean Value 74114. Properties of the Mean Value 74215. Variance 74416*. Correlation 74617. Characteristic Functions 748§ 4. Applications of the Normal Law 75018. The Normal Law as the Limiting One 75019. Confidence Interval 75220. Data Processing 754CHAPTER XIX. COMPUTERS 757§ 1. Two Classes of Computers 7571. Analogue Computers 7582. Digital Computers 762§ 2. Programming 7643. Number Systems 7644. Representing Numbers in a Computer 7665. Instructions 7696. Examples of Programming 772Appendix. Equations of Mathematical Physics 7801*. Derivation of Some Equations 7802*. Some Other Equations 7833*. Initial and Boundary Conditions 784§ 2. Method of Separation of Variables 7864*. Basic Example 7865*. Some Other Problems 791Bibliography 796Name Index 798Subject Index 8OOList of Symbols 81