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# Real Functions in Several Variables: Volume III

### Differentiable Functions in Several Variables

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Language:  English
The topic of this series of books on "Real Functions in Several Variables" is very important in the description in e.g. Mechanics of the real 3-dimensional world that we live in.
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The topic of this series of books on "Real Functions in Several Variables" is very important in the description in e.g. Mechanics of the real 3-dimensional world that we live in. Therefore, we start from the beginning, modelling this world by using the coordinates of R3 to describe e.b. a motion in space.

The theory and methods of these volumes on "Real Functions in Several Variables" are applied constantly in higher Mathematics, Mechanics and Engineering Sciences. It is of paramount importance for the calculations in Probability Theory, where one constantly integrate over some point set in space.

It is my hope that this text, these guidelines and 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.

1. Preface
2. Introduction to volume III, Differentiable Functions in Several Variables
3. Differentiable functions in several variables
1. Differentiability
2. The chain rule
3. Directional derivative
4. Cn-functions
5. Taylor’s formula
4. Some useful procedures
1. Introduction
2. The chain rule
3. Calculation of the directional derivative
4. Approximating polynomials
5. Examples of differentiable functions
2. The chain rule
3. Directional derivative
4. Partial derivatives of higher order
5. Taylor’s formula for functions of several variables
6. Formulæ
1. Squares etc
2. Powers etc
3. Differentiation
4. Special derivatives
5. Integration
6. Special antiderivatives
7. Trigonometric formulæ
8. Hyperbolic formulæ
9. Complex transformation formulæ
10. Taylor expansions
11. Magnitudes of functions
7. Index

Leif Mejlbro

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.