Real Functions of Several Variables - Applicati...

Book Calculus 2c-9 - Real Functions in Several Variables - Examples of Applications of Gauß’s and Stokes’s Theorems and Related Topics

In this volume I present some examples of applications of Gauß’s and Stokes’s theorems and related topics,, cf. also Calculus 2b, Functions of Several Variables. Since my aim also has been to demonstrate some solution strategy I have as far as possible structured the examples according to the following form

A Awareness, i.e. a short description of what is the problem.

D Decision, i.e. a reflection over what should be done with the problem.

I Implementation, i.e. where all the calculations are made.

C Control, i.e. a test of the result.

This is an ideal form of a general procedure of solution. It can be used in any situation and it is not linked to Mathematics alone. I learned it many years ago in the Theory of Telecommunication in a situation which did not contain Mathematics at all. The student is recommended to use it also in other disciplines.

One is used to from high school immediately to proceed to I. Implementation. However, examples and problems at university level are often so complicated that it in general will be a good investment also to spend some time on the first two points above in order to be absolutely certain of what to do in a particular case. Note that the first three points, ADI, can always be performed.

This is unfortunately not the case with C Control, because it from now on may be difficult, if possible, to check one’s solution. It is only an extra securing whenever it is possible, but we cannot include it always in our solution form above.

It is my hope that these examples, of which many are treated in more ways to show that the solutions procedures are not unique, may be of some inspiration for the students who have just started their studies at the universities.

Finally, even if I have tried to write as careful as possible, I doubt that all errors have been removed. I hope that the reader will forgive me the unavoidable errors.

Leif Mejlbro

Preface

1. Gradient elds and integrals

2. The flux of a vector eld

3. The divergence and rotation of a vector eld

4. Gauβ’s theorem

5. Stokes’s theorem

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.