A Guide Book to Mathematics

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 Nystrom 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.