Real Functions of Several Variables - Max, Vect...

A book concerning Calculus 2c - Real Functions in Several Variables, Examples of Maximum and Minimum Integration and Vector Analysis

The purpose of this volume is to present some worked out examples from the theory of Functions in Several Variables in the following topics:

1) Maximum and minimum of a function.

2) Integration in the plane and in the space.

3) Vector analysis.

As an experiment I shall here use the following generic diagram for solving problems:

A. For Awareness. What is the problem?

Try to formulate the problem in your own words, thereby identifying it.

D. For Decision. What are we going to do with it?

Are there any reasonable solution procedure available? If so, which one should be chosen?

I. For Implementation. Here we do all the necessary calculations for solving the task after the choice of the previous D.

At high school one usually starts here, but the problems may now be so complex that we need the previous analysis as well.

C. For Control. Whenever it is possible, one should check the solution.

Note, however, that this is not always possible, so in many cases we have to skip this point.

Notice that A, D, I can always be effectuated, no matter whether the problem is a mathematical exercise, or construction of some building, or any other problem which should be solved. The model is in this sense generic. It was first presented for me in Telecommunication for over 15 years ago, where I added C, the control of the solution. I hope that these simple guidelines will help the students as much as it has helped me. Notice also that if one during the I, Implementation, comes across a new and unforeseen problem, then one may iterate this simple model.

The intension is not to write a textbook, but only 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 are only consisting of examples without any further mathematical theory, which one must get from an ordinary textbook. On the other hand, it should be possible to copy the methods given here in similar exercises.

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 and of Calculus 2b, Real Functions in Several Variables, Methods of Solution.

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

1. Preface

2. The range of a function in several variables
2.1 Maximum and minimum
2.2 Extremum

3. The plane integral
3.1 Rectangular coordinates
3.2 Polar coordinates

4. The space integral
4.1 Rectangular coordinates
4.2 Semi-polar coordinates
4.3 Spherical coordinates

5. The line integral

6. The surface integral

7. Transformation theorems

8. Improper integrals

9. Vector analysis
9.1 Tangential line integral; gradient field
9.2 Flux and divergence of a vector field; Gauss’s theorem
9.3 Rotation of a vector field; Stokes’s theorem
9.4 Potentials

A Formulæ
A.1 Squares etc.
A.2 Powers etc.
A.3 Differentiation
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

Leif Mejlbro was educated as a mathematician at the University of Copenhagen, where he wrote his thesis on Linear Partial Differential Operators and Distributions. Shortly after he obtained a position at the Technical University of Denmark, where he remained until his retirement in 2003. He has twice been on leave, first time one year at the Swedish Academy, Stockholm, and second time at the Copenhagen Telephone Company, now part of the Danish Telecommunication Company, in both places doing research.

At the Technical University of Denmark he has during more than three decades given lectures in such various mathematical subjects as Elementary Calculus, Complex Functions Theory, Functional Analysis, Laplace Transform, Special Functions, Probability Theory and Distribution Theory, as well as some courses where Calculus and various Engineering Sciences were merged into a bigger course, where the lecturers had to cooperate in spite of their different background. He has written textbooks to many of the above courses.

His research in Measure Theory and Complex Functions Theory is too advanced to be of interest for more than just a few specialist, so it is not mentioned here. It must, however, be admitted that the philosophy of Measure Theory has deeply in uenced his thinking also in all the other mathematical topics mentioned above.

After he retired he has been working as a consultant for engineering companies { at the latest for the Femern Belt Consortium, setting up some models for chloride penetration into concrete and giving some easy solution procedures for these models which can be applied straightforward without being an expert in Mathematics. Also, he has written a series of books on some of the topics mentioned above for the publisher Ventus/Bookboon.