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 X, Vector fields; Gauß’s Theorem
3. Tangential line integrals
   1. Introduction
   2. The tangential line integral. Gradient fields
   3. Tangential line integrals in Physics
   4. Overview of the theorems and methods concerning tangential line integrals and gradient fields
   5. Examples of tangential line integrals
4. Flux and divergence of a vector field. Gauß’s theorem
   1. Flux
   2. Divergence and Gauß’s theorem
   3. Applications in Physics
   4. Procedures for flux and divergence of a vector field; Gauß’s theorem
   5. Examples of flux and divergence of a vector field; Gauß’s theorem
5. 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
6. Index