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Real Functions in Several Variables: Volume XI

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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.

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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 XI, Vector fields II; Stokes’s Theorem; nabla calculus
  3. Rotation of a vector field; Stokes’s theorem
    1. Rotation of a vector field in R3
    2. Stokes’s theorem
    3. Maxwell’s equations
    4. Procedure for the calculation of the rotation of a vector field and applications of Stokes’s theorem
    5. Examples of the calculation of the rotation of a vector field and applications of Stokes’s theorem
  4. Nabla calculus
    1. The vectorial differential operator ?
    2. Differentiation of products
    3. Differentiation of second order
    4. Nabla applied on x
    5. The integral theorems
    6. Partial integration
    7. Overview of Nabla calculus
    8. Overview of partial integration in higher dimensions
    9. Examples in nabla calculus
  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
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