Content

Introduction

Lecture ¹1. The functions of many variables and their properties. The limits and continuity 4

Lecture ¹2. The partial derivatives and differentials of functions of many variables 9

Lecture ¹3. The higher-order partial derivatives and differentials. Taylor’s formula 18

Lecture ¹4. The existence and the differentiability of the implicit function. The implicit functions defined in system of functional equations 23

Lecture ¹5. The maximum and minimum of function depending from many variables 28

Lecture ¹6. The double integral and its properties. The geometric meaning. The calculation of double integrals 34

Lecture ¹7. The change of the variable in the double integrals. Practical application of double integrals 39

Lecture ¹8. The triple integral and its properties. The geometric meaning. The calculation of triple integrals 42

Lecture ¹9. The change of the variable in the triple integrals. Practical application of triple integrals 44

Lecture ¹10. Improper Integrals Depending on a Parameter. Improper Multiple Integrals 47

Lecture ¹11. The curvilinear integrals 50

Lecture ¹12. Green's Theorem. Applications of line integrals. 54

Lecture ¹13. The integrals over the surface 56

Lecture ¹14. The elements of a field‘s theory: 59

Lecture ¹15. Vector interpretation Stoks’s formula. Potential field; vector line and vector pipes; solenoid field; “nabla” operator 62

List of literature 65