Antiderivatives and Plane Integrals

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166 pages

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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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*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 h...

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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.

- Preface
- Introduction to volume VI, Integration of a function in several variables
- Antiderivatives of functions in several variables
- The theory of antiderivatives of functions in several variables
- Templates for gradient fields and antiderivatives of functions in three variables
- Examples of gradient fields and antiderivatives

- Integration in the plane
- An overview of integration in the plane and in the space
- Introduction
- The plane integral in rectangular coordinates
- Examples of the plane integral in rectangular coordinates
- The plane integral in polar coordinates
- Procedure of reduction of the plane integral; polar version
- Examples of the plane integral in polar coordinates
- Examples of area in polar coordinates

- Formulæ
- Squares etc
- Powers etc
- Differentiation
- Special derivatives
- Integration
- Special antiderivatives
- Trigonometric formulæ
- Hyperbolic formulæ
- Complex transformation formulæ
- Taylor expansions
- Magnitudes of functions

- Index