Fundamentals of Tensor Calculus for Engineers with a Primer on Smooth Manifolds

134 Pages·2017·1.425 MB·English

by  Uwe Mühlich

Solid Mechanics and Its Applications
Uwe Mühlich
Fundamentals of
Tensor Calculus for
Engineers with a
Primer on Smooth
Manifolds
Solid Mechanics and Its Applications
Volume 230
Series editors
J.R. Barber, Ann Arbor, USA
Anders Klarbring, Linköping, Sweden
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ü
Uwe M hlich
Fundamentals of Tensor
Calculus for Engineers
with a Primer on Smooth
Manifolds
123
UweMühlich
Faculty of AppliedEngineering
University of Antwerp
Antwerp
Belgium
ISSN 0925-0042 ISSN 2214-7764 (electronic)
Solid MechanicsandIts Applications
ISBN978-3-319-56263-6 ISBN978-3-319-56264-3 (eBook)
DOI 10.1007/978-3-319-56264-3
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Preface
Continuum mechanics and its respective subtopics such as strength of materials,
theoryofelasticity,andplasticityareofutmostimportanceformechanicalandcivil
engineers. Tensors of different types, such as vectors and forms, appear most
naturallyinthiscontext.Ultimately,whenitcomestolargeinelasticdeformations,
operations like push-forward, pull-back, covariant derivative, and Lie derivative
become inevitable. The latter form a part of modern differential geometry, also
known as tensor calculus on differentiable manifolds.
Unfortunately, in many academic institutions, an engineering education still
relies on conventional vector calculus and concepts like dual vector space, and
exterioralgebraaresuccessfullyignored.Theexpression“manifold”arisesmoreor
less as a fancy but rather diffuse technical term. Analysis on manifolds is only
mastered and applied by a very limited number of engineers. However, the mani-
foldconcepthasbeenestablishednowfordecades,notonlyinphysicsbut,atleast
in parts, also in certain disciplines of structural mechanics like theory of shells.
Over the years, this has caused a large gap between the knowledge provided to
engineering students and the knowledge required to master the challenges, con-
tinuum mechanics faces today.
The objective of this book is to decrease this gap. But, as the title already
indicates, it does not aim to give a comprehensive introduction to smooth mani-
folds. On the contrary, at most it opens the door by presenting fundamental
concepts of analysis in Euclidean space in a way which makes the transition to
smooth manifolds as natural as possible.
The book is based on the lecture notes of an elective course on tensor calculus
taughtatTU-BergakademieFreiberg.Theaudienceconsistedofmasterstudentsin
Mechanical Engineering and Computational Materials Science, as well as doctoral
students of the Faculty of Mechanical, Process and Energy Engineering. This
introductory text has a special focus on those aspects of which a thorough under-
standing is crucial for applying tensor calculus safely in Euclidean space, partic-
ularly for understanding the very essence of the manifold concept. Mathematical
proofs are omitted not only because they are beyond the scope of the book but
also because the author is an engineer and not a mathematician. Only in some
v
vi Preface
particularcasesareproofssketchedinordertoraiseawarenessoftheeffortmadeby
mathematicians to work out the tools we are using today. In most cases, however,
theinterestedreaderisreferredtocorrespondingliterature.Furthermore,invariants,
isotropictensorfunctions,etc.,arenotdiscussed,sincethesesubjectscanbefound
in many standard textbooks on continuum mechanics or tensor calculus.
Prior knowledge in real analysis, i.e., analysis in R, is assumed. Furthermore,
students should have a prior education in undergraduate engineering mechanics,
including statics and strength of materials. The latter is surely helpful for under-
standing the differences between the traditional and modern version of tensor
calculus.
Antwerp, Belgium Uwe Mühlich
February 2017
Acknowledgements
Iwouldliketoexpressmygratitudetomyformercolleaguesofthesolidmechanics
group of the Institute of Mechanics and Fluid Dynamics at TU-Bergakademie
Freiberg, and in particular to Michael Budnitzki, Thomas Linse, and Christoph
Häusler. Without their open-minded interest in the subject and their constant
readiness todiscussit,this bookwouldnotexist. Iamespecially grateful toHeinz
Gründemann for his support and inspiring discussions about specific topics related
tothesubject.Inthiscontext,IwouldalsoliketothankJarnoHein,whobroughtus
into contact.
Additionally,thisbookcouldnothavebeenaccomplishedwithouttheeducation
I received in engineering mechanics, and I would like to take this opportunity to
express my sincere gratitude to my teachers, Reinhold Kienzler and Wolfgang
Brocks.
IwouldliketothankTU-BergakademieFreibergandespeciallyMeinhardKuna
for the opportunity to teach a course about tensor calculus and for providing the
freedom to teach a continuum mechanics lecture during the summer term of 2014
which deviated significantly from the approach commonly applied in engineering
education. In addition, I would like to express my appreciation to all students and
colleagues who attended the courses mentioned above for their interest and aca-
demic curiosity, but especially for critical remarks which helped to uncover mis-
takes and to clarify doubts.
Besides, I would like to acknowledge the support provided by the Faculty of
Applied Engineering of the University of Antwerp in terms of infrastructure and
time necessary to finalize the process of converting the course notes into a book.
Furthermore,IwouldliketothankNathalieJacobs,AlmitraGosh,NagaKumar
Natti, Henry Pravin, and Cynthia Feenstra from Springer Scientific Publishing for
their support throughout the process of publishing this book. My thanks are also
due to Marc Beschler for language editing the book.
vii
Contents
1 Introduction.... .... .... ..... .... .... .... .... .... ..... .... 1
1.1 Space, Geometry, and Linear Algebra .... .... .... ..... .... 1
1.2 Vectors as Geometrical Objects . .... .... .... .... ..... .... 2
1.3 Differentiable Manifolds: First Contact.... .... .... ..... .... 3
1.4 Digression on Notation and Mappings .... .... .... ..... .... 6
References.. .... .... .... ..... .... .... .... .... .... ..... .... 8
2 Notes on Point Set Topology.... .... .... .... .... .... ..... .... 9
2.1 Preliminary Remarks and Basic Concepts.. .... .... ..... .... 9
2.2 Topology in Metric Spaces. .... .... .... .... .... ..... .... 10
2.3 Topological Space: Definition and Basic Notions.... ..... .... 15
2.4 Connectedness, Compactness, and Separability.. .... ..... .... 17
2.5 Product Spaces and Product Topologies... .... .... ..... .... 19
2.6 Further Reading .... ..... .... .... .... .... .... ..... .... 21
References.. .... .... .... ..... .... .... .... .... .... ..... .... 22
3 The Finite-Dimensional Real Vector Space.... .... .... ..... .... 23
3.1 Definitions .... .... ..... .... .... .... .... .... ..... .... 23
3.2 Linear Independence and Basis.. .... .... .... .... ..... .... 25
3.3 Some Common Examples for Vector Spaces ... .... ..... .... 28
3.4 Change of Basis.... ..... .... .... .... .... .... ..... .... 29
3.5 Linear Mappings Between Vector Spaces.. .... .... ..... .... 30
3.6 Linear Forms and the Dual Vector Space.. .... .... ..... .... 32
3.7 The Inner Product, Norm, and Metric. .... .... .... ..... .... 35
3.8 The Reciprocal Basis and Its Relations with the Dual Basis. .... 37
References.. .... .... .... ..... .... .... .... .... .... ..... .... 40
4 Tensor Algebra . .... .... ..... .... .... .... .... .... ..... .... 41
4.1 Tensors and Multi-linear Forms . .... .... .... .... ..... .... 41
4.2 Dyadic Product and Tensor Product Spaces .... .... ..... .... 43
4.3 The Dual of a Linear Mapping.. .... .... .... .... ..... .... 46
ix
x Contents
4.4 Remarks on Notation and Inner Product Operations.. ..... .... 46
4.5 The Exterior Product and Alternating Multi-linear Forms... .... 48
4.6 Symmetric and Skew-Symmetric Tensors.. .... .... ..... .... 50
4.7 Generalized Kronecker Symbol . .... .... .... .... ..... .... 51
4.8 The Spaces KkV and KkV(cid:2). .... .... .... .... .... ..... .... 52
4.9 Properties of the Exterior Product and the Star-Operator ... .... 53
4.10 Relation with Classical Linear Algebra.... .... .... ..... .... 55
References.. .... .... .... ..... .... .... .... .... .... ..... .... 57
5 Affine Space and Euclidean Space... .... .... .... .... ..... .... 59
5.1 Definitions and Basic Notions .. .... .... .... .... ..... .... 59
5.2 Alternative Definition of an Affine Space
by Hybrid Addition . ..... .... .... .... .... .... ..... .... 61
5.3 Affine Mappings, Coordinate Charts
and Topological Aspects... .... .... .... .... .... ..... .... 62
References.. .... .... .... ..... .... .... .... .... .... ..... .... 67
6 Tensor Analysis in Euclidean Space.. .... .... .... .... ..... .... 69
6.1 Differentiability in R and Related Concepts Briefly Revised .... 69
6.2 Generalization of the Concept of Differentiability.... ..... .... 72
6.3 Gradient of a Scalar Field and Related Concepts in RN .... .... 73
6.4 Differentiability in Euclidean Space Supposing
Affine Relations .... ..... .... .... .... .... .... ..... .... 77
6.5 Characteristic Features of Nonlinear Chart Relations . ..... .... 84
6.6 Partial Derivatives as Vectors and Tangent Space at a Point .... 86
6.7 Curvilinear Coordinates and Covariant Derivative ... ..... .... 89
6.8 Differential Forms in RN and Integration .. .... .... ..... .... 93
6.9 Exterior Derivative and Stokes’ Theorem in Form Language.... 95
References.. .... .... .... ..... .... .... .... .... .... ..... .... 98
7 A Primer on Smooth Manifolds. .... .... .... .... .... ..... .... 99
7.1 Introduction ... .... ..... .... .... .... .... .... ..... .... 99
7.2 Basic Concepts Regarding Analysis on Surfaces in R3..... .... 103
7.3 Transition to Smooth Manifolds. .... .... .... .... ..... .... 108
7.4 Tangent Bundle and Vector Fields... .... .... .... ..... .... 109
7.5 Flow of Vector Fields and the Lie Derivative... .... ..... .... 114
7.6 Outlook and Further Reading... .... .... .... .... ..... .... 118
References.. .... .... .... ..... .... .... .... .... .... ..... .... 119
Solutions for Selected Problems.... .... .... .... .... .... ..... .... 121
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 123