principles and applications of tensor analysis - m. smith

BASIC TENSOR THEORY of relative tensors, covariant tensors, contra- variant tensors, mixed tensors, metric tensors, base vectors, and Kronecker deltas.. The equations for the two-bod

$6.95 Cat No TAS-1

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Principles and Applications of Tensor Anal-

ysis presents a detailed step-by-step develop-

Ai, f

concepts in tensor analysis, differential geom-

etry, and analytical mechanics in tensor form

CHAPTER 1 BASIC TENSOR THEORY

of relative tensors, covariant tensors, contra-

variant tensors, mixed tensors, metric tensors, base vectors, and Kronecker deltas

ordinates Some of the more important applica-

‘ons in potential t wacluded_as ill

tive examples

$6.95 Cat No TAS-1

Principles & Applications

of Tensor Analysis

Copyright © 1963 by Howard —¥ Sams _&—_Co n

Indianapolis 6, Indiana Printed in the United States

of America

editorial or pictorial content, in any manner, is pro-

hibited No patent liability is assumed with respect to

the use of the information contained herein

Library of Congress Catalog Card Number: 63-11937

Since the extensive use of tensors by Einstein, important applications in other fields, such as differential geometry, classical mechanics, and the theory of elasticity, have evolved Therefore, the ability to understand and apply tensors is a definite advantage to engineers, physicists, and applied math-

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give added insight to the understanding 0 e fundamen

laws of physics and engineering

This text, usi ical les tÌ hout, s

atically explains and demonstrates basic tensor theory and its applications The book is divided into four chapters The first two chapters present basic tensor theory, including Christoffel symbols, covariant derivatives, and Laplace’s

equation The third chapter includes the Riemann-Christoffel

, and t Heati tị Iwsie 4 ;

topics in differential geometry The latter were selected be- cause they have numerous important applications in the gen-

of elasticity

The fourth chapter presents the application of tensor

analysis to some of the most important concepts in classical

and relativistic mecnanics € section on Classical dynamics

includes Lagrange’s equations of motion and a sol

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two-body problems The equations for the two-body problem

are written as a geodesic to give the student a feeling for

dynamical trajectories treated as geodesics prior to the study

The sections on relativistic mechanics include a discussion

of space-time for the special theory of relativity, the Lorentz

It is hoped that an understanding of the material in this

text wi vide th t with th ili

of problems using tensor analysis, as well as giving him an additional understanding of the related sciences

CUM ATT EW S SMITH February, 1963

Contents

CHAPTER I

BASIC TENSOR TTHEORY . 2S 222 S22 SE SE=S SE SE==Ss========-

Summation Notation—Relative Tensors—Admissible Trans-

formations—-N Dimensional Space—Contravariant Tensors—

Covariant Tensors— Higher Rank and Mixed Tensors—Metric

Tensors and the Line Element-— Base Vectors — Associated

Tensors and the Inner Product—Kronecker Deltas

CHAPTER II

s 7 1 a L2

CC? RLS FOR EE VMBOLS AND CL A OV ARLAN EF V2 y Lu =< F)

Christoffel Symbols—Transformation of Christoffel Symbols—

Covariant Derivatives—Higher Rank and Mixed Covariant

Derivatives—Intrinsic Derivatives—Laplace’s Equation

CHAPTER WE _RIEMANN-CHRISTOFFEL- TENSORS &

DIFFERENTIAL GEOMETRY . -.-. -eeceee-ceneeeeeeeeceeneeeeeceee 69

Riemann-Christoffel Tensors—Ricci Tensors—Gaussian Curva-

ture—Serret-Frenet Formulas—-Geodesics—Parallel Displace-

ment—Surfaces, First Fundamental Form—Surfaces, Second

Fundamental Form

CHAPTER IV

_— MECHANICS ẦĐB - CLASSICAL AND RELATIVISTIC

Dynamics of a Particle for Classical Mechanics—Lagrange’s

quations o otion for Classica echanics—Classical Solu-

tion of the Two-Body Problem—Geodesic Equations for the

Two-Body Problem in Classical Mechanics—Minkowski Space- Time and the Lorentz Transformations — Four-Dimensional

Minkowski Momentum Vector, and Hinstein’s Energy Equation Minkowski Acceleration and Force Vectors — Hinstein’s - ravitati 1 ivity—

Relativistic Solution of the Two-Body Problem for the General Theory of Relativity

Chapter 1©

Basic Tensor Theory

Chapter Ï

Basic Tensor Theory

T lysis is the study of invariant objeets, wÌ

properties must be independent of the co-ordinate systems used to describe the objects A tensor is represented by a set

of functions called components For an object to be a tensor

it must be an invariant that transforms from one acceptable

several examples of tensors are velocity vectors, base

« vectors, metric coefficients for the length of a line, Gaussian curvature, and the Newtonian gravitation potential

Many of the important di a i 1

in tensor form are Lagrange’s equations of motion and Laplace’s equation When an equation is written in tensor form it is in a general form that applies to all admissible co-ordinate systems

_ a

TENSOR ANALYSIS

Cố 7 šSUMMATION NOTATION ——=—

The summation notation used throughout this text will

be demonstrated explicitly by the examples in the first chapter The general summation will be of the type:

vy For additional simplification, Einstein dropped the ye + 7 s s ° e es

in Equation 1.1 and the summation is then expressed:

S —

Gtx er

In conclusion, it is to be remembered that Equations 1.0, 1.1, and 1.2 are equivalent, and xi expresses variables, not powers of x

In the subsequent portions of the text, a superscript index

will indicate a contravariant tensor, while a subscript index

will indicate a covariant tensor

The rank of a tensor is the sum of the covariant and contravariant indexes This will be explained in detail in the section of this chapter on higher rank and mixed tensors

In cartesian co-ordinates (x, y) it-is:

The Jacobian of the cartesian co-ordinates with respect to

H 1 -ordinates is f if the followi Ha]

r r

ing Jacobian:

Now, returning to the expression for differential area in

cartesian co-ordinates and polar co-ordinates, the following

equation can be written: co

S dx dy = S dr do

where,

S=1

S=r

Exponent n in Equations 1.5 and 1.6 is used to determine

* ° ° ° Awe Ate Ay AMA Aya Kh NH ^®ÐNGu@ an 4 are Ona *

(2 V r3) - iJ

n — 1 An absolute scalar has a weight of zero; i.e, n = 0

To illustrate Equation 1.5, we use the values:

yi ranges from i = 1 toi = 2

xi ranges from j =1toj = 2

Equation 1.7 is the desired result

Now the notion of relative tensors can be extended to vol- umes and mass To illustrate this concept, we start with the equation for an incremental mass in orthogonal cartesian co-

Now the incremental mass in spherical co-ordinates is writ-

ten in terms of relative tensor S:

18

BASIC TENSOR THEORY

The geometrical relationship between the cartesian co-or- dinates and the spherical co-ordinates is indicated in Fig 1-2

TENSOR ANALYSIS

Using these values, the resultant determinate is:

sin@sin® rcos@sin® rsin#cos@|=r?sin# (1.11)

COS & —r sin ® O

_Now, Equation 1.6 can be evaluated: —

S = r'sin&p (1.12)

KO Oo

BASIC TENSOR THEORY

dM = pr’sin & dr dé do (1.18)

Equation 1.13 is the desired result for dM If the value for

dM had been given initially in spherical co-ordinates, the cor-

responding value in cartesian co-ordinates could be found by _the equation: C77777

relative tensors transform from one co-ordinate system to an-

other by means of the functional determinate known as the Jacobian Since a relative tensor is defined to be a function of

"he jacoDlan a necessa and sulicient condition Tor an ad-

of a set in which the Jacobian does not vanish

This condition is also necessary and sufficient for absolute tensors Therefore, the set of all admissible transformations

of co-ordinates form a group with nonvanishing Jacobians

an admissible transformation of co-ordinates can be expressed:

TENSOR ANALYSIS

An example of Equation 1.16 can be found in Jacobian Equa- — - tions 1.3 and 1.4

In general terms a co-ordinate system represents a one-to-

one correspondence of a point or object with a set of numbers

To measure distance, we can use a rectangular cartesian co-

ordinate system This is called a metric manifold, or space of V

Now, a space or manifold of N dimensions is expressed by

point or object

A sub space Vu where M = N - 1 is called a hypersurface

An example of hypersurface in Euclidean space is a plane It

is a hypersurface of V.,

CONTRAVARIANT TENSORS

The prototype for contravariant tensors is the vector

formed by taking the total differential of a variable in one co-

BASIC TENSOR THEORY

To illustrate Equation 1.22, we write the corresponding

tensor notation for Equation 1.20

dx

=a (1.23)

bị

For x = rcos®, ar = cos 0, and— = -rsin@

Equation 1.20 can be expressed:

BASIC TENSOR THEORY

With 4} lues, { on Bi = , H

oxz following result:

vector formed by taking the partial derivative of an absolute

scalar The chain rule for partial derivatives gives the correct ' or this { Fị f Hon Tk j ‘ant Ten-

TENSOR ANALYSIS

Âm ÌỲ— — — ct eee C— X_" oe sen mmợẾP

PL

Fig 1-3

u Mm (Length) * v=- r vf (Force) (Time) 4 (1.31)

Force F in the direction of the x axis is:

N nợ the vai oV „Mm or

0 Now, using the values 2t 1m” and ax > cos ® in Equation 1.34 yi :

26

BASIC TENSOR THEORY

TENSOR ANALYSIS

ÒXxz — OX? oO OX" A

— ap oy! oy!

Dij

The equation for a contravariant tensor, Rank 2, is:

Ox* = ox? AK A af “~~ ft = ;_ “wes”

The equation for mixed tensor, Rank 2, is:

b9 oo

BASIC TENSOR THEORY

Metric tensors ø;; of the line element ds? are covariant

tensors, Rank 2 To present the concept of the metric tensors, line element ds is written as a vector in cartesian and polar co-ordinates Using Fig 1-4 as a reference, line element ds in

iss — i idx? + 2i+jaxdy + j*¢jay? (44)

Using an orthogonal co-ordinate system and elementary vector analysis, the dot products are:

Therefore, the square of the line element is:

Coefficients g,,, 2 and g,,, Z are covariant tensors, Rank 2

To illustrate the transformation of the metric tensors, we will use Equation 1.40 for the transformation of covariant -

BASIC TENSOR THEORY

OX Ox _ sY oy

Soe — QO Qo Sur TT ò® ò@ Zoe (1.53)

= = cos 0, — = sin ®

or or

2, = cos?@ + sin?@ = 1 (1.57)

Using the computed values for g,, and g,,, as indicated in

Equations 1.57 and 1.54, we get the familiar result for the

ds? = dr’? + r’do? (1.58)

TỊ ‘ant ric 4 Rank 2; led 4 ‘ant

tensor is called the contravariant fundamental tensor For orthogonal co-ordinate systems, the contravariant metric ten-

sor is found by the equation:

3]

(1.41)

To illustrate the procedure, the metric tensors from the fol-

lowing equation are used:

BASIC TENSOR THEORY

This is the desired result

The determinate of the metric tensors is denoted by sym- bol g For Euclidian space, it is:

~ ~e ~ ~ (3 @QO

6 — | S21 S22 E22 \L.02)

231 Boo 23 _If the co-ordinate system is orthogonal, the determinate re-

duees to;

ø„, 0 0

g=—!0 øg,, 0 (1.63)

0 0 go, The general formula to find the contravariant metric tensors 1S?

Gi

=

£

G4 is a cofactor of elements gii in the determinate g

For all of the examples in this text, orthogonal co-ordinate

systems are used This simplifies the evaluation of the metric

tensors and the tensor equations = eee

set of orthogonal cartesian co-ordinates x, y, and z, with unit

vectors i, j, and k, and position vector P are indicated in Fig 1-5

33

TENSOR ANALYSIS

The corresponding base vectors are:

œ rg

I ey |

BASIC TENSOR THEORY

Now, the same position vector P expressed in cylindrical co- ordinates r, ©, and z is:

B | ‘ant 4 Rank 1 To illustrat

this relation e, in Equations 1.66 will be found in terms of

e, = (icos® + jsin®) cos®

+ (-irsin © + jr cos ®) (-—— sin @)

C On

TENSOR ANALYSIS

The metric tensors can be expressed as the dot product of

the base vectors To illustrate the procedure, the following

Hither the product of a covariant metric tensor and a con- travariant tensor, or a contravariant metric tensor and a co-

variant tensor are called associated tensors An example of

21a 4 L) 4 ava ¬Ï]| ê are nh ava

25 É at dì u

ector To illustrate this fact, polar co-ordinates r

and ©, and cartesian co-ordinates x and y are used In the

polar co-ordinates, the metric tensors and covariant base vec-

tors are:

BASIC TENSOR THEORY

Using Equation 1.76 and the values in Equations 1.75, base

vectors e' and e? are:

TENSOR ANALYSIS

———— From Equation 1.76 and the following calculations it can — ~

be seen that the contravariant metric tensor, Rank 2, and the covariant base vector, Rank 1, form a contravariant base vec- tor or tensor, Rank 1 Now, the corresponding equation for a

| n-Fouati 1.75,-t] luct g g” — 1 wil ]

calculated to demonstrate that ø,, ø! is a mixed tensor, co- variant Rank 1, and contravariant Rank 1

C3 oo

BASIC TENSOR THEORY KRONECKER DELTAS —— ———

An example of a Kronecker Delta is the equation:

Ox* oyl Se ‘oy k (1.88)

Next, we let k = j = 2, andi = 1, 2

2

O sin? @ + cos? 0 = 1

39

TENSOR ANALYSIS

~ ‘The Kronecker Delta is a mixed tensor, Rank 2, whose com- ˆ

ponents in any other admissible co-ordinate system again form

a Kronecker Delta

Chapter Hi

Christoffel Symbols and the Covariant

Derivat;