A Guide Book to Mathematics

TO THE FIRST RUSSIAN EDITION It was a very difficult task to write a guide-book of a small size designed to contain the fundamental knowledge of mathema­ tics which is most necessary to engineers and students of higher technical schools. In our tendency to the compactness and brevity of the exposition, we attempted, however, to produce a guide-book which would be easy to understand, convenient to use and as accurate as possible (as much as it is required in engineering). It should be pointed out that this book is neither a handbook nor a compendium, but a guide-book. Therefore it is not written as systematically as a handbook should be written. Hence the reader should not be surprised to find, for example, I'HOpital's rule in the section devoted to computation of limits which is a part of the chapter "Introduction to the analysis" placed before the concept of the derivative, or information about the Gamma function in the chapter "Algebra"-just after the concept of the factorial. There are many such "imperfections" in the book. Thus a reader who wants to acquire certain information is advised to use not only the table of contents but also the alpha­ betical index inserted at the end of the book. If a problem mentioned in the text is explained in detail in another place of the book, then the corresponding page is indicated in a footnote.

one Tables and Graphs.- I. Tables.- A. Tables of elementary functions.- 1. Some frequently occurring constants.- 2. Squares, cubes, roots.- 3. Powers of integers from n = 1 to n = 100.- 4. Reciprocals of numbers.- 5. Factorials and their reciprocals.- 6. Some powers of the numbers 2, 3 and 5.- 7. Common logarithms.- 8. Antilogarithms.- 9. Natural values of trigonometric functions.- 10. Exponential, hyperbolic and trigonometric functions (for x from 0 to 1.6).- 11. Exponential functions (continued) (for x from 1.6 to 10).- 12. Natural logarithms.- 13. Length of circumference of a circle with diameter d.- 14. Area of a circle with diameter d.- 15. Elements of the segment of the circle.- 16. Sexagesimal measure of angles expressed in radians.- 17. Proportional parts.- 18. Table of quadratic interpolation.- B. Tables of special functions.- 19. The Gamma function.- 20. Bessel’s cylindrical functions.- 21. Legendre’s polynomials.- 22. Elliptic integrals.- 23. Probability integral.- II. Graphs.- A. Elementary functions.- 1. Polynomials.- 2. Rational functions.- 3. Irrational functions.- 4. Exponential and logarithmic functions.- 5. Trigonometric functions.- 6. Inverse trigonometric functions.- 7. Hyperbolic functions.- 8. Inverse hyperbolic functions.- B. Important curves.- 9. Curves of the third degree.- 10. Curves of the fourth degree.- 11. Cycloids.- 12. Spirals.- 13. Some other curves.- two Elementary Mathematics.- I. Approximate computations.- 1. Rules of approximate computations.- 2. Approximate formulas.- 3. Slide rule.- II. Algebra.- A. Identity transformations.- 1. Fundamental notions.- 2. Integral rational expressions.- 3. Rational fractional expressions.- 4. Irrational expressions; transformations of exponents and radicals.- 5. Exponential and logarithmic expressions.- B. Equations.- 6. Transformation of algebraic equations into canonical form.- 7. Equations of the first, second, third and fourth degree.- 8. Equations of the n-th degree.- 9. Transcendental equations.- 10. Determinants.- 11. Solution of a system of linear equations.- 12. System of equations of higher degrees.- C. Supplementary sections of algebra.- 13. Inequalities.- 14. Progressions, finite series and mean values.- 15. Factorial and gamma function.- 16. Variations, permutations, combinations.- 17. Newton’s binomial theorem.- III. Geometry.- A. Plane geometry.- 1. Plane figures.- B. Solid geometry.- 2. Straight lines and planes in space.- 3. Angles in space.- 4. Polyhedrons.- 5. Curvilinear solids.- IV. Trigonometry.- A. Plane trigonometry.- 1. Trigonometric functions.- 2. Fundamental formulas of trigonometry.- 3. Harmonic quantities.- 4. Solution of triangles.- 5. Inverse trigonometric functions.- B. Spherical trigonometry.- 6. Geometry on a sphere.- 7. Solution of spherical triangles.- C. Hyperbolic trigonometry.- 8. Hyperbolic functions.- 9. Fundamental formulas of hyperbolic trigonometry.- 10. Inverse hyperbolic functions.- 11. Geometric definition of hyperbolic functions.- three Analytic and Differential Geometry.- I. Analytic geometry.- A. Geometry in the plane.- 1. Fundamental concepts and formulas.- 2. Straight line.- 3. Circle.- 4. Ellipse.- 5. Hyperbola.- 6. Parabola.- 7. Curves of the second degree (conic sections).- B. Geometry in space.- 8. Fundamental concepts and formulas.- 9. Plane and straight line in space.- 10. Surfaces of the second degree (canonical equations).- 11. Surfaces of the second degree (general theory).- II. Differential geometry.- A. Plane curves.- 1. Ways in which a curve can be defined.- 2. Local elements of a curve.- 3. Points of special types.- 4. Asymptotes.- 5. General examining of a curve by its equation.- 6. Evolutes and involutes.- 7. Envelope of a family of curves.- B. Space curves.- 8. Ways in which a curve can be defined.- 9. Moving trihedral.- 10. Curvature and torsion.- C. Surfaces.- 11. Ways in which a surface can be defined.- 12. Tangent plane and normal.- 13. Linear element of a surface.- 14. Curvature of a surface.- 15. Ruled and developable surfaces.- 16. Geodesic lines on a surface.- four Foundations of Mathematical Analysis.- I. Introduction to analysis.- 1. Real numbers.- 2. Sequences and their limits.- 3. Functions of one variable.- 4. Limit of a function.- 5. Infinitesimals.- 6. Continuity and points of discontinuity of functions.- 7. Functions of several variables.- 8. Series of numbers.- 9. Series of functions.- II. Differential calculus.- 1. Fundamental concepts.- 2. Technique of differentiation.- 3. Change of variables in differential expressions.- 4. Main theorems of differential calculus.- 5. Finding maxima and minima.- 6. Expansion of a function into a power series.- III. Integral calculus.- A. Indefinite integrals.- 1. Fundamental concepts and theorems.- 2. General rules of integration.- 3. Integration of rational functions.- 4. Integration of irrational functions.- 5. Integration of trigonometric functions.- 6. Integration of other transcendental functions.- 7. Tables of indefinite integrals.- B. Definite integrals.- 8. Fundamental concepts and theorems.- 9. Evaluation of definite integrals.- 10. Applications of definite integrals.- 11. Improper integrals.- 12. Integrals depending on a parameter.- 13. Tables of certain definite integrals.- C. Line, multiple and surface integrals.- 14. Line integrals of the first type.- 15. Line integrals of the second type.- 16. Double and triple integrals.- 17. Evaluation of multiple integrals.- 18. Applications of multiple integrals.- 19. Surface integrals of the first type.- 20. Surface integrals of the second type.- 21. Formulas of Stokes, Green and Gauss-Ostrogradsky.- IV. Differential equations.- 1. General concepts.- A. Ordinary differential equations.- 2. Equations of the first order.- 3. Equations of higher orders and systems of equations.- 4. Solution of linear differential equations with constant coefficients.- 5. Systems of linear differential equations with constant coefficients.- 6. Operational method of solution of differential equations.- 7. Linear equations of the second order.- 8. Boundary-value problems.- B. Partial differential equations.- 9. Equations of the first order.- 10. Linear equations of the second order.- five Supplementary Chapters on Analysis.- I. Complex numbers and functions of a complex variable.- 1. Fundamental concepts.- 2. Algebraic operations with complex numbers.- 3. Elementary transcendental functions.- 4. Equations of curves in complex form.- 5. Functions of a complex variable.- 6. Simplest conformal mappings.- 7. Integrals in the domain of complex numbers.- 8. Expansion of analytic functions into power series.- II. Vector calculus.- A. Vector algebra and vector functions of a scalar.- 1. Fundamental concepts.- 2. Multiplication of vectors.- 3. Covariant and contravariant coordinates of a vector.- 4. Geometric applications of vector algebra.- 5. Vector function of a scalar variable.- B. Field theory.- 6. Scalar field.- 7. Vector field.- 8. Gradient.- 9. Line integral and potential in a vector field.- 10. Surface integral.- 11. Space differentiation.- 12. Divergence of a vector field.- 13. Rotation of a vector field.- 14. The operators ? (Hamilton’s operator), (a?) and ? (Laplace’s operator).- 15. Integral theorems.- 16. Irrotational and solenoidal vector fields.- 17. Laplace’s and Poisson’s equations.- III. The calculus of variations.- 1. Fundamental principles.- 2. The simple variation problem with one unknown function.- 3. Sufficient conditions for the assumption of an extremum.- 4. The variation problem in polar coordinates.- 5. The inverse problem of the variational calculus.- 6. The variation problem in parametric form.- 7. Base functions involving derivatives of higher orders.- 8. The Euler differential equations for the variation problem with n unknown functions.- 9. The extremum of a multiple integral.- 10. The variation problem with side conditions.- 11. The isoperimetric problem of tho calculus of variations.- 12. Two geometric variation problems with two independent variables.- 13. Ritz’s method of solution of variation problems.- IV. Integral equations.- 1. General notions.- 2. Simple integral equations which can be reduced to ordinary differential equations by differentiation.- 3. Integral equations which can be solved by differentiation.- 4. The Abel integral equation.- 5. Integral equations with product kernels.- 6. The Neumann series (successive approximation).- 7. The method of solution of Fredholm.- 8. The Nyström method of approximation for the solution of Fredholm integral equations of the second kind.- 9. The Fredholm alternative theorem for Fredholm integral equations of the second kind with symmetric kernel.- 10. The operator method in the theory of integral equations.- 11. The Schmidt series.- V. Fourier series.- 1. General information.- 2. Table of certain Fourier expansions.- 3. Approximate harmonic analysis.- six Interpretation of Experimental Results.- I. Foundations of the theory of probability and the theory of errors.- 1. Theory of probability.- 2. Theory of errors.- II. Empirical formulas and interpolation.- 1. Approximate representation of a functional dependence.- 2. Parabolic interpolation.- 3. Selection of empirical formulas.