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Vector and tensor practice. 2nd revised ed. (Vektor- und Tensorpraxis.) (German) Zbl 1175.00013

Frankfurt am Main: Harri Deutsch (ISBN 978-3-8171-1837-3/pbk). x, 308 p. (2009).
The book covers all aspects of vector algebra and vector calculus in three dimensions which seem relevant to students of physics and engineering sciences. The presentation favors intuitive understanding over the rigorous development of an axiomatic theory. Nonetheless, all results are proved and all calculations are carried out explicitly. Vectors and tensors in spaces of higher dimensions are only treated cursorily.
The author motivates theoretical results by numerous examples taken from electrical engineering, mechanical engineering, physics, astronomy, geometry, etc. Each of the eleven chapters contains an exercise section whose solutions are given in the appendix. Examples as well as exercises contain many particular results of interest (for example the solutions to the isoperimetric problem and the brachistochrone problem or a derivation of Snell’s law of refraction) whose proofs use exclusively the machinery developed within the book.
Throughout the text historical remarks are interspersed. The book contains a useful index but no bibliography or suggestions for further reading. The German chapter and section titles and their English translations are
1.
Vektoralgebra (Vector Algebra)
1.1
Grundbegriffe (Basic Concepts)
1.2
Elementare Operationen (Elementary Operations)
2.
Produkte von Vektoren (Vector Products)
2.1
Punktprodukt (Dot Product)
2.2
Kreuzprodukt (Cross Product)
2.3
Elementare Vektorgleichungen
(Elementary Vector Equations)
2.4
Spatprodukt (Triple Product)
2.5
Mehrfache Kreuzprodukte
(Multiple Cross Products)
3.
Analytische Geometrie (Analytic Geometry)
3.1
Gerade (Straight Line)
3.2
Ebene (Plane)
3.3
Kegelschnitte (Conic Sections)
3.4
Lineare Transformationen kartesischer Koordinaten (Linear Transformations of Cartesian Coordinates)
4.
Vektoranalysis (Vector Calculus)
4.1
Einführung (Introduction)
4.2
Örtliche Differenzialoperationen 1. Ordnung (Differential Operations of First Order with Respect to Position)
4.3
Anwendungen in Mathematik und Physik (Applications to Mathematics and Physics)
4.4
Örtliche Differenzialoperationen 2. Ordnung (Differential Operations of Second Order with Respect to Position)
5.
Differenzialgeometrie (Differential Geometry)
5.1
Räumliche Kurven und Bahnen (Spatial Curves and Trajectories)
5.2
Krumme Flächen (Curved Surfaces)
6.
Krummlinige rechtwinkelige Koordinaten \(u\), \(v\), \(w\) (Curvilinear Orthogonal Coordinates \(u\), \(v\), \(w\))
6.1
Transformation von \(x\), \(y\), \(z\) zu \(u\), \(v\), \(w\) (Transformation from \(x\), \(y\), \(z\) to \(u\), \(v\), \(w\))
6.2
Spezielle Koordinaten (Special Coordinates)
7.
Vektorielle Integrale (Vectorial Integrals)
7.1
Grundregeln (Fundamental rules)
7.2
Linien- und Umlaufintegrale (Line and Contour Integrals)
7.3
Flächen- und Hüllenintegrale (Surface and Hull Integrals)
7.4
Schwerpunkte (Barycenters)
8.
Integralsätze (Integral Theorems)
8.1
Satz von Stokes (Stokes Theorem)
8.2
Satz von Gauß (Gauß’ Theorem)
8.3
Formel von Gauß (Gauß’ Formula)
8.4
Satz von Green (Green’s Theorem)
8.5
Singularitäten im Integrationsgebiet (Singularities in the Domain of Integration)
8.6
Greensche Funktion (Green’s Funktion)
9.
Parameterintegrale (Parameter Integrals)
9.1
Linienintegrale (Line Integrals)
9.2
Flächenintegrale (Surface Integrals)
9.3
Raumintegrale (Volume Integrals)
10.
Variationsrechnung (Calculus of Variation)
10.1
Geschichte (History)
10.2
Probleme ohne Nebenbedingungen
(Problems Without Side Conditions)
10.3
Probleme mit Nebenbedingungen
(Problems With Side Conditions)
11.
Tensorrechnung (Tensor Calculus)
11.1
Grundlagen (Fundamentals)
11.2
Wechsel des Bezugssystems
(Change of the Reference Frame)
11.3
Orthogonale Transformationen
(Orthogonal Transformations)
11.4
Eigensystem (Eigensystem)
A.
Lösung der Aufgaben (Solutions to the Exercises)
B.
onkurrierende Begriffe (Competing Concepts)

MSC:

00A06 Mathematics for nonmathematicians (engineering, social sciences, etc.)
15A69 Multilinear algebra, tensor calculus
53A45 Differential geometric aspects in vector and tensor analysis
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