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Selected Sequences of Russian Mathematics Textbooks Translation
Preface to the original book
The first part is the differential calculus of unary function.
Chapter 1 Introduction
The first lecture
The operation of the 1. collection. Cartesian product of a set. Maps and functions.
The second speech
2. Potential of continuum of countable and uncountable sets of equivalent sets.
The third lecture
3. Real numbers
The fourth lecture
4. Completeness of real number set
55. Lemmas on separability of sets, nested interval systems and contraction closures.
Lemma of interval sequence
Chapter II Limits of Sequence of Numbers
The fifth lecture
1. Mathematical induction, Newton binomial and Bernoulli inequality
2. Sequence, infinitesimal sequence and infinite sequence and their properties
Lecture 6
3. Limit of sequence.
4. Limit process in inequality
Lecture 7
5. Monotone sequence. Wilstrass theorem. The number "e" and Euler constant.
Lecture 8
6. Porzano-Wilstrass Theorem on the existence of partial limit of bounded sequence.
7. Cauchy criterion of sequence convergence
Chapter III Limit of Function at One Point
Lecture 9
1. The concept of numerical function limit
2. Set the foundation. Limit along the basis function
Lecture 10
3. Take the limit of inequality as an example
4. Cauchy criterion for the existence limit of functions along the basis
Lecture 11
5. Equivalence between Cauchy convergence definition and Heine convergence definition.
6. Theorem about the limit of composite function
7. Order of infinitesimal function
Chapter IV Continuity of Functions at a Certain Point
Lecture 12
1. The Properties of Functions Continuous at One Point
2. Continuity of elementary functions
Lecture 13
3. Important restrictions
4. Continuity of functions on a set
Lecture 14
5. General properties of continuous functions on closed intervals
Lecture 15
6. The concepts of consistency and continuity.
7. Properties of closed sets and open sets. Compactness. Continuous functions on compact sets
The fifth chapter is the differential of unary function.
Lecture 16
The increment of 1. function. Differential and derivative of function
Lecture 17
2. Differential of compound function
3. Differential law
Lecture 18
4. Higher derivative and higher differential
5. A little function increase or decrease
Lecture 19
6. Rolle Theorem, Cauchy Theorem and Lagrange Theorem.
Lecture 20
7. Inference of Lagrange theorem.
8. Some inequalities
9. Derivative of a function given as a parameter
Lecture 2 1
10. Extension of infinitive
Lecture 22
1 1. local Taylor formula
12. Taylor formula with general remainder
Lecture 23
13. The application of Taylor formula in some functions
Lecture 24
14. Learn the function with the help of derivative. Convexity of extreme point
Lecture 25
15. inflection point
Lecture 26
16. Insert text
Lecture 27
17. Secant method and tangent method (Newton method). Fast calculation
Chapter VI Indefinite Integral
Lecture 28
1. primitive function. integrable function
Lecture 29
2. The nature of indefinite integral
Lecture 30
Supplement. The generalization of the limit concept of Heine method to the function that converges along the set basis
The second part is the differential calculus of Riemann integral multivariate function.
Chapter VII Definite Integral
Chapter 8 Basic Theorem of Riemann Integral Theory
Chapter 9 improper integral
Chapter 10 The Length of Curve
Chapter 1 1 Jordanian measures
Chapter 12 Lebesgue measure theory and Lebesgue integral theory. stieltjes integral
Chapter 13 Some concepts of general topology. metric space
Chapter 14 Differential calculus of multivariate functions
Part III Function Series and Parameter Integral
Chapter 15 Numerical Sequence
Chapter 16 Function Sequence and Function Series
Chapter 17 Integral Dependent on Parameters
Chapter 18 Fourier Series and Fourier Integral
The fourth part is the surface integral of multiple Riemann integrals.
Chapter 19 Multiple integrals
Chapter 20 Surface Integral
Chapter 21 General Stokes Formula