Vectors and tensors in engineering and Physics

300 Pages·2·13.429 MB·English

by  Danielson, D. A.

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Vectors and Tensors
in Engineering and Physics,
SECOND EDITION
D. A. Danielson
Naval Postgraduate School
esv iCW Advanced Book Program
A Member of the Perseus Books Group
In memory of my mother and father
All rights reserved. Printed in the United States of America. No part of this publication
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Copyright © 2003 by Donald A. Danielson
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ISBN 0-8133-4080-2
The paper used in this publication meets the requirements of the American National
Standard for Permanence of Paper for Printed Library Materials Z39.48-1984.
Text designed and typeset by the author
10 9 8 i 6 5
CONTENTS
PREFACE
VECTOR ALGEBRA '
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ee
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TeO me TYol G eta ATU aIN Le VEL OTS snakeeee e G
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TENSOR ALGEBRA
Cee CEN OO Har) EEN Ene Serre nt hr at Seen Lt, ans Gig?
BDO EOP GEL Ls Oba WN oO] metic cec c, 4 geet g <b iee -i
2.3 MORE PROPERTIES OF SECOND-ORDER TENSORS .....
Dam EVok SAE Rely)to n) Semen hr RAN a Ope Bega Bee ie ST
Demme CAE EOIN) BH GIS ee ee Be ne Sih Rote malian uaa 5
CARTESIAN COMPONENTS
3.1 COMPONENTS OF A VECTOR AND
RORC Es OPAL OTN Ui ia ot Roca ats Bee
been RoI) Ve ONL re ce tan a eee ee des
oe IGEN VEC LOR ALEPREGEN LATIONS spike vetpees euvur «6
3.4 GEOMETRIC DISTORTIONS AND THE POLAR
DECOM POS UPA SCH OEM earns ee eae eee
TPE TAR LD. tA UL ALON S ode em ER Tet etcuis Grace Gighae S
OMEN ELC Pe d GELS re cei 1 Oe eres eecM Ess aotmt a Meo e tans
GENERAL COMPONENTS
4.1 CHANGE OF CARTESIAN BASE VECTORS AND
COORDINATE Vo LE Wo touareg ene Oe me ttn ei cect >
Ae a Ne bee WO-ODIMENSIONS 2 ec ikealaah g e
ao eenO LATIONS INSTHREE “DIMENSIONS = ) seeSteen c e
44 INVARIANCE OF TENSOR COMPONENTS ...........
Aaa a BN ORAS Aco bite) ns ee oe ee ee EERE 6S
ADEN LAL MINGOR OTHERS 80 ve 5 co wots 0. mere REC 5.a eo se
TENSOR FIELDS OF ONE VARIABLE
pee BASIC CALCULUS OF VECTOR PIBLDS@ nesess oo
5 meV ING DA SEV EO LORS @ ouies ntiicenet tie Glave RLMEa hae ws
5.3 NEWTONIAN MECHANICS OF A PARTICLE ..........
5.4 DIFFERENTIAL GEOMETRY OF A SPACE CURVE ......
ome MO LLON CO teAthiIGLLN BOLY s a erocle& be aj he ene ar
OO Le LONGO TsO LAE RS meee ess bet. ko piney os
iv Contents
6 TENSOR FIELDS OF MANY VARIABLES 99
6.1 DIFFERENTIATION OF SCALAR FIELDS ............ 99
6.2 DIFFERENTIATION OF VECTOR FIELDS............ 104
6.3 LINE, SURFACE, AND VOLUME INTEGRALS.......... 108
6.4 GREEN’S THEOREM AND POTENTIAL FIELDS OF
TWOSVARIA BLES ere ee ee ee 113
6.5 STOKES’ AND DIVERGENCE THEOREMS AND POTENTIAL
FIELDS OF THREE VARIABLES ............... ren ee
6.GaeNOTATIONIOFSOTHERS 0-00. aren ee a ae 129
7 APPLICATIONS 131
Tal SHEATACONDUGCTION teat yearn aea e ee 131
72m SOLID MECHANICS mete. ee nee 138
73s ELUID MECHANICS = eee See re 148
7.4. NEWTONIAN ORBITAL MECHANICS .............. 157
7.5 ELECTROMAGNETIC THEORY ...... Pe ee 164
7.6 NOTATION OF OTHERS .......... 5 a eae Bee Pee 176
8 GENERAL COORDINATES
177
8.1 GENERAL CURVILINEAR COORDINATES............ 177
8.2 ORTHOGONAL CURVILINEAR COORDINATES ........ 184
5 3 SURFACE COORDINATES 0-3 uu ee a 191_
8.4 MECHANICS OF CURVED MEMBRANES ............ 200
8S NOTATION OF OTR ERS anes weg ee ee ae 205
9 FOUR-DIMENSIONAL SPACETIME 207
OSU ECLA LAE ALINGL V on ag 207
DEMEGE NECA LTR DA LIVI)Y ancy oe 216
JromeNaO LILON tORL OTH ERS gaara ne 222
REFERENCES
225
HISLTORICAL E ee Set en a ee ee ee Poke I a 225
MATHEMATICAL 2 00/on 0et e2e2 225
PHYSICAL ee te ene ee ee Seen eee ee 227
ANSWERS TO PROBLEMS
231
INDEX 275
PREFACE
I have written this text for those who want to understand the close correspondence
between mathematics and the physical world. The emphasis herein on physical
interpretation means that certain aspects of mathematical rigor are neglected.
“Rigor” to a mathematician can be “rigor mortis” to an engineer.
An advanced undergraduate student majoring in a physical science, engineer-
ing, or mathematics would have the necessary prerequisites for this book. The
reader is expected to know only some elementary calculus and linear algebra. In
order to make the subject accessible to a wide audience, certain valuable mathe-
matical tools such as the calculus of variations have not been used.
The focus of revision for this second edition is the insertion of additional steps
in the derivations of mathematical formulas and problem solutions. The changes
will make it easier for the reader to learn the subject matter solely from this text.
Whereas my own students were the proving ground for the material in the first
edition, other readers have suggested many of the improvements in this second
edition. My heartfelt thanks goes to each of these individuals who seek the truth.
The universe stands continually open to our gaze, but it cannot be understood
unless one first learns to comprehend the language and interpret the characters in
which it ts written. It is written in the language of mathematics, and tts characters
are triangles, circles, and other geometrical figures, without which it 1s humanly
impossible to understand a single word of it; without these, one is wandering about
in a dark labyrinth.
— Galileo Galilei, 1623
Chapter 1
VECTOR ALGEBRA
1.1 INTRODUCTION
Tensor mathematics is a beautiful, simple, and useful language for the description
of natural phenomena. Tensor fields are the abstract symbols of this language.
Each tensor field represents a single physical quantity that is associated with
certain places in three-dimensional space and instants of time.
Quantities such as the mass of a satellite, the temperature at points in a body,
and the charge of an electron have a definite magnitude. They can be represented
adequately by pure numbers or scalars (tensors of order zero). Properties such
as the position or velocity of a satellite, the flow of heat in a body, and the
electromagnetic force on an electron have both magnitude and direction. They
are better represented by directed line segments or vectors (tensors of order one).
Other quantities such as the stress inside a solid or fluid may be characterized by
tensors of order two or higher.
Tensor equations express the relationship between physical quantities. We
will study the basic equations governing the trajectories of point masses in a
gravitational field, motion of finite rigid bodies, transfer of heat by conduction,
deformation of solids, flow of fluids, and electromagnetic phenomena. By solving
these equations, we will be able to predict the course of future events.
You will gain from this book a manipulative skill with tensors and an ability
to apply them to model our physical world. Then when you encounter a tensor
in your studies or research, you will recognize it as such and have the correct
mathematical tools to work with.
1.2 HISTORICAL NOTES
Great scientific breakthroughs result from the accumulated efforts of many re-
searchers working over a long period of time. Nevertheless, later generations in-
accurately attribute these achievements solely to just one or two individuals who
first assembled the ideas of their predecessors into a unifying theory or book. We
list here some of the individuals who are now credited with the development of
our field. For an exciting experience in the world of mathematics, read the classic
works under historical references at the back of this book.
Euclidean geometry is based on the Elements by the Greek Euclid (300 B.C.).
Cartesian coordinates are named after the French scientist Descartes (1596-1650).
Newton (1642-1727) presented the motion of the planets and other bodies in
strictly geometrical terms. The great mathematician Euler (1707-1783) used
Cartesian components for force and moment of inertia. The French mathematician
2 1 Vector Algebra
Cauchy (1789-1857) correctly described the general state of stress in an elastic
body. The words “scalar,” “vector,” and “tensor”! were used by the Irish mathe-
matician Hamilton (1805-1865). The German mathematicians Gauss (1777-1855)
and Riemann (1826-1866) were major contributors to the metrical geometry of
non-Euclidean manifolds. The great physicist Maxwell (1831-1879) was aware
that the dielectric permittivity, the magnetic permeability, and the conductivity
for electric currents may be linear vector operators. Our present vector and dyadic
notation is almost the same as that used in the old book by the Americans Gibbs
and Wilson (1901). Tensor calculus was perfected by the Italian mathematicians
Ricci (1853-1925) and Levi-Civita (1873-1941). Einstein (1879-1955) gets credit
for first applying the generalized calculus of tensors to gravitation.
1.3. VECTOR BASICS
In the first erght chapters of this book, we model physical space by three-dimensional
Euchdean geometry. In a Euclidean space we may construct straight line segments.
If we ascribe a direction to a line segment, it becomes a vector. A vector may be
represented geometrically by an arrow (see Fig. 1.1). The direction of a vector is
the direction of its arrow, and the magnitude of a vector is the length of its arrow.
We represent a vector algebraically by a lowercase boldface letter, such as v, e,
or f.
A scalar is simply a pure number. In the text of this book every number is real;
that is, every number can be represented by a terminating or nonterminating dec-
imal. Scalars are designated by italic letters, such as c, k, or W. The magnitude
(length) of a vector v is denoted by |v| or v. Of course, the magnitude (abso-
lute value) of a scalar c is also denoted by |c|. (Both |v| and |c| are nonnegative
ee | 20
Figure 1.1. The sum and difference of vectors u and v; multiplication of v by the
scalars 2 and 0.
1The word “scalar” stems from the Latin “scalae” meaning “stairs” (scale of numbers from
negative to positive infinity). The word “vector” stems from the Latin “vectus” meaning
“carried” (denoting a direction). The word “tensor” stems from the Latin “tensus” meaning
“stretched” (tension).