Real Functions of Several Variables - Guidelines..

A book concerning Calculus 2b - Real Functions in Several Variables.

The purpose of this volume is to give some guidelines for the student concerning the solution of problems in the theory of Functions in Several Variables.

The intension is not to write a textbook, but instead to give some hints of how to solve problems in this field. It therefore cannot replace any given textbook, but it may be used as a supplement to such a book on Functions in Several Variables.

The chapters have been kept as short as possible, each dealing with one particular type of standard problem, supplied with short descriptions of the possible methods of solutions which may be available, their advantages and also their shortcomings.

In Appendix A the reader will find a collection of formulæ which otherwise tacitly are assumed to be known from high school. It is highly recommended that the student learns these by heart during the course, because they form the backbone of the elementary part of Calculus, which should be mastered, before one may proceed to more advanced parts of Mathematics.

The text is a continuation of Calculus 1a, Real Functions in One Variable. It will be followed by a volume called Calculus 2c, in which selected examples from the theory of functions in several variables are considered.

The text is based on my experiences in my teaching of students in this course. I realized that there was absolutely a need for a practical description of how to solve explicit problems.

Leif Mejlbro

Preface

1. Integration of trigonometric polynomials

2. Complex decomposition of a fraction between two polynomials

3. Integration of a fraction between two polynomials

4. The domain of a function

5. The chain rule

6. Calculation of the directional derivative

7. Approximating polynomials

8. Gradient elds; antiderivatives

9. An overview of integration in the plane and in the space

10. Reduction of a plane integral; rectangular version

11. Reduction of a plane integral; polar version

12. Reduction of a space integral; rectangular version

13. Reduction of a space integral; semi-polar version

14. Reduction of a space integral; spherical version

15. Line integrals

16. Surface integrals

17. Transformation theorems

18. Improper integrals; bounded domain

19. Improper integrals; unbounded domain

20. The setting up of a line, a plane, a surface or a space integral

21. Tangential line integral; gradient eld

22. Flux and divergence of a vector eld; Gau ’s theorem

23. Rotation of a vector eld; Stokes’s theorem

24. Green’s theorems in the plane

25. Nabla calculus

26. Partial integration in higher dimensions

27. Vector potential

28. Global maximum and minimum

29. Extrema; local investigation

A. Formulæ
A.1 Squares etc.
A.2 Powers etc.
A.3 Dierentiation
A.4 Special derivatives
A.5 Integration
A.6 Special antiderivatives
A.7 Trigonometric formulæ
A.8 Hyperbolic formulæ
A.9 Complex transformation formulæ
A.10 Taylor expansions
A.11 Magnitudes of functions