E. Byskov, Elementary Continuum Mechanics for Everyone,Solid Mechanics and Its Applications 194, DOI: 10.1007/978-94-007-5766-0_31,Ó Springer Science+Business Media Dordrecht 2013

Chapter 31

Index Notation, theSummation Convention,and a Little About TensorAnalysisIn most cases it proves too cumbersome to write all terms of the equations Index Notation

Summation

Convention

which represent some physical phenomenon. Therefore, some shorthand no-tation is preferable. One of these is the so-called Index Notation, which isoften useful, in particular when it is combined with the Summation Con-vention, see below.

In the present chapter we give an introduction to the subject of the in-dex notation and the summation convention, but postpone many of thedetails to the sections on continuum mechanics, e.g. Section 2.2, whichseems to be a more natural place to introduce the definitions of severaltopics.

31.1 Index NotationIn a three-dimensional Cartesian31.1 coordinate system the axes are often Cartesian

Coordinate Systemdenoted the x-, y- and z-axes. This is rather inconvenient for many purposesas we shall see shortly. And, by the way, at the same time the base vectors ofthe (x, y, z) coordinate system are often called i1, i2, i3, respectively, whichdoes not seem consistent. Furthermore, if we wish to indicate that a scalarvalued function f depends on all three coordinates we must write f(x, y, x)which for one thing is not very elegant and for another rather lengthy.

In terms of its components the divergence div v of a vector field v may

31.1 In Cartesian coordinate systems the axes are straight lines, which are at right anglesto each other, and the unit measure along the axes is the same and constantly independentof the position of the point in question. Although the axes of a polar coordinate systemare at right angles, it is not a Cartesian coordinate system because one of the familiesof coordinate lines consists of circles, while the other is formed by straight lines. As aconsequence, the length measure varies with the distance from the pole.

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be written

Divergence divv div v =∂vx∂x

+∂vy∂y

+∂vz∂z

(31.1)

which is longer than necessary in that the structure of all terms is thesame. Therefore, we seek a notation which is compact and yet displays thecontents of the equations in a reasonably decipherable form. Here, the IndexIndex Notation

Notation proves to be of good help.

The first step is to replace x by x1, y by x2, and z by x3. The nextstep is to introduce a counter, the index i, i ∈ [1, 2, 3]. Then31.2 we mayLowercase Roman

subscripts take the

values 1, 2, 3

refer to the above-mentioned scalar function f as f(xi), or f(xk) for thatmatter because we let all lowercase Roman subscripts (lower indices) takethe values 1, 2, 3. With this index notation in hand we may rewrite (31.1)

Divergence divv div v =∂v1∂x1

+∂v2∂x2

+∂v3∂x3

(31.2)

which, however, is even longer and more complicated than (31.1). We shalltherefore introduce further simplifications, see Sections 31.3 and 31.2.

31.2 Comma NotationAnother convenient shorthand notation is the convention that comma indi-cates partial differentiation. Thus,

Comma notation

( ),j( ),j ≡

∂( )

∂xj(31.3)

which means that we may write (31.2) in a somewhat shorter form

Divergence divv div v = v1,1 + v2,2 + v3,3 (31.4)

31.3 Summation ConventionFirst, note that we may write vi instead of v to indicate the vector. Then,Roman indices:

range [1, 3] introduce the Summation Convention, which states that a repeated low-ercase Roman index indicates a sum from 1 to 3 over the index and mustappear twice—not three or four times—in each term of an expression. Then,e.g.

Summation

Convention: Sum

over repeated

lower-case index

xixi ≡3∑

i=1

(xi)2 or xjxj ≡

3∑

k=1

(xk)2 (31.5)

where repeated indices, here i and j in (31.5a) and (31.5b), respectively, aresummation indices. Such indices are also called a dummy indices becauseSummation index

Dummy index their names are irrelevant.31.2 In Section 31.3.1 we introduce lowercase Greek indices to cover the values 1, 2.

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Now we can write (31.1) in a very compact and convenient form

Divergence divvdiv v =∂vi∂xi

(31.6)

or, even shorter

Divergence divvdiv v = vi,i (31.7)

There is a self-evident result, which we shall utilize time and again, namely

∂xi∂xj

= δij (31.8)

where δij denotes the Kronecker delta which is defined by

Kronecker delta

δijδij ≡

{1 for j = i

0 for j 6= i(31.9)

The Kronecker delta may be used to change index of some quantityUse the Kronecker

delta to change

index

vi = δijvj (31.10)

see also (2.3) and the derivation following that equation.

For the Kronecker delta δij the following formula is useful and followsfrom the summation convention

δjj = 3δjj = 3 (31.11)

Another useful symbol is the Permutation Symbol eijk, whose definitionin the 3-dimensional case is31.3

Permutation

symbol eijkeijk ≡

0 if any two of the subscripts are equal

+1 if (i, j, k) = (1, 2, 3) or (2, 3, 1) or (3, 1, 2)

−1 if (i, j, k) = (3, 2, 1) or (2, 1, 3) or (1, 3, 2)

(31.12)

The vector product of two vectors a and b is

Vector product

v = a× bv = a× b (31.13)

which, by use of the permutation symbol the vector product, may be writtenin component form.

Vector product

vi = eijkajbkvi = eijkajbk (31.14)

Here, index i takes the values 1, 2 and 3 resulting in the three expressions

31.3 Do not confuse the permutation symbol with a linear strain measure. As mentionedbefore, the many different notations for strain cause problems.

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for v1, v2 and v3, respectively. Thus, while j and k are dummy indices in(31.14) i plays another role and is called a called a free index.31.4Free index

In some cases we consider the curl, also called the rotation, of a vectorfield and in this connection the permutation symbol proves to be useful.The curl of a vector field v with components vj is defined by

Curl v curl v ≡

∣∣∣∣∣∣∣∣∣

i1 i2 i3

∂

∂x1

∂

∂x2

∂

∂x3v1 v2 v3

∣∣∣∣∣∣∣∣∣(31.15)

In components this may be written

Curl v ∼ ci curl(vj) = −eijkvj,k (31.16)

where ci denotes the components of the curl of vk.

31.3.1 Lowercase Greek Indices

Sometimes we shall deal with two-dimensional bodies such as plates, andin that connection it is convenient to be able to distinguish between thesummation convention for two- and three-dimensional bodies. For two-Lowercase Greek

indices: range [1, 2] dimensional bodies it is customary to let Greek lowercase letters play thesame role as the lowercase Roman letters for the three-dimensional bodies.

The two-dimensional equivalent of (31.5) clearly isSummation

Convention: Sum

over repeated

lower-case Greek

index

xαxα ≡2∑

α=1

(xα)2 or xβxβ ≡

2∑

γ=1

(xγ)2 (31.17)

and the two-dimensional version of (31.11) is

δαα = 2 δαα = 2 (31.18)

The obvious definition of the Permutation Symbol eαβ for the two-dimensional case is31.5

Two-dimensional

case: Permutation

symbol eαβ

eαβ ≡

0 for (α, β) = (1, 1) or (α, β) = (2, 2)

+1 for (α, β) = (1, 2)

−1 for (α, β) = (2, 1)

(31.19)

and thus it is seen from (31.12) and (31.19) that the Permutation Symbolmay be generalized to cases with more indices in that its value is always 0when two indices are equal, while it is +1 when the indices are (1, 2, . . .) oran even permutation thereof, and −1 otherwise.

31.4 In a way the term free index is a little misleading in that i must be the same on bothsides of the equation, while j and k could be substituted by n and m without alteringthe value of vi.31.5 Also in the two-dimensional case there is a problem regarding notation in that eαβ

might be confused with the linear strain in a plate.

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31.3.2 Symmetric and Antisymmetric Quantities

31.3.2.1 Product of a Symmetric and an Antisymmetric Matrix

For our immediate purpose we consider a matrix Ajk31.6 which is symmetric

in its indices

Symmetric matrixAkj = Ajk (31.20)

and another matrix Bjk which is antisymmetric31.7

Antisymmetric

matrixBkj = −Bjk (31.21)

and try to compute the value of their inner product AjkBjk

AjkBjk = 12AjkBjk + 1

2AkjBkj =12AjkBjk + 1

2AjkBkj (31.22)

where in the second term we have interchanged dummy indices.

Then, The inner product

between a

symmetric and an

antisymmetric

matrix i zero

AjkBjk = 12Ajk(Bjk +Bkj) = 0 (31.23)

because of (31.21). We have then shown that the inner product between asymmetric and an antisymmetric matrix vanishes.

31.3.2.2 Product of a Symmetric and a General Matrix

The above finding has an important side effect. Given two matrices, namelyAjk which is symmetric and another. Bmn which may be of a more generalcharacter in that it is neither symmetric, nor antisymmetric. Then we maysplit Bmn in a symmetric part BS

mn = BSnm and an antisymmetric part

BAmn = −BA

nm

Bmn = BSmn +BA

mn (31.24)

The inner product of Ajk and Bmn then isIn the inner

product between

a symmetric and

general matrix the

antisymmetric

part is wiped out

AmnBmn = Amn(BSmn +BA

mn) = AmnBSmn (31.25)

because AmnBAmn = 0 according to the above result. Thus, the inner prod-

uct between a symmetric and a general matrix does not contain any infor-mation about the antisymmetric part of the general matrix.

31.6 Actually, the following derivation is valid for matrices as well as (Cartesian) tensors,see page 544.31.7 The term antisymmetric is somewhat funny in that symmetric means “with themetric” and, therefore, antisymmetric must be “against with the metric.” The termantimetric, which is used in some European countries therefore makes more sense.

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31.3.3 Summation Convention Results in Brevity

Let us consider the following expression—don’t try to interpret it

aα = bαζωcζdω (31.26)

and observe that the summation convention is indeed a very handy tool,because written out (31.26) not only signifies two equations, but also adouble sum, i.e. 4 terms on the right-hand side of each equation. Had theindices been Roman instead of Greek, the number of equations would havebeen 3 and the number of terms in each equation 9 instead of 4.

In Section 2.6 and Chapter 5 we encounter formulas like

σij = Eijklεkl (31.27)

which is exceedingly lengthy if written out in full, and it becomes obviousthat there is a vast saving in terms of writing effort by using the indexnotation in conjunction with the summation convention—without sacrificingthe possibility to identify each individual term of the equations. Let us pickthe expression for σ13 as an example. Using the summation convention

σ13 = E13jkεjk (31.28)

and in full

σ13 =+ E1311ε11 + E1312ε12 + E1313ε13

+ E1321ε21 + E1322ε22 + E1323ε23

+ E1331ε31 + E1332ε32 + E1333ε33

(31.29)

There are 9 of these expressions, so the amount of space taken up bywriting (31.27) out in full is so large that you—at least I—very easily loseperspective.

Tensor Analysis

It seems fair to mention that the subjects covered in this chapter are veryTensor Analysis is

much more than we

cover here

specialized examples of Tensor Analysis, which deals with the description ofquantities in any coordinate system, e.g. curvilinear coordinate systems. Inthe present context of an introduction to continuum mechanics we do notneed more than Cartesian Tensors, i.e. quantities in Cartesian coordinatesystems, see Section 31.1.

31.4 Generalized CoordinatesThe concept of a coordinate is conveniently broadened to cover vectors andfunctions.

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31.4.1 Vectors as Generalized Coordinates

The Cartesian coordinate system may be said to be spanned by its base Vectors as

Generalized

Coordinates

vectors ij , see Section 2.2.1. In the same spirit we may use any set oflinearly independent vectors, such as jk, k ∈ [1, N ], as the basis for anN -dimensional space.31.8 We may simply choose jk such that its only non-vanishing component is the kth, which is taken to be 1. Then any vectorin the N -dimensional space may be written as a sum of its components,i.e. coordinates, in the direction of the N base vectors just as any vectorin the three-dimensional space may be resolved in terms of the three basevectors ij . Some people find it helpful to think in this way, while others findit confusing. The latter may as well proceed without further speculationsalong this line.

For the sake of introducing vectors as generalized coordinates, considerthe following problem: We may wish to know how the length of a vector vvaries with respect to one of its components, say vj . For mathematical easewe shall consider the square of the length instead of the length itself31.9

Functional ΠΠ(v) = Π(vk) = vivi (31.30)

where Π designates the square of the length.31.10 Then, differentiation withrespect to vj provides

Partial derivative

with respect to a

vector

∂Π(vi)

∂vj= 2

∂vi∂vj

vi (31.31)

Obviously, we need to be able to compute the value of ∂vi/∂vj. This isvery easy when we realize that the only meaningful rule is

∂vi∂vj

≡{1 for j = i

0 for j 6= i(31.32)

The right-hand side of (31.32) is the Kronecker delta δij , which wasdefined in (31.9) and in Section 2.2, i.e.

∂vi∂vj

= δij (31.33)

Application of the definition of the Kronecker delta makes it possible towrite (31.31) as

∂Π(vi)

∂vj= 2δijvi = 2vj (31.34)

31.8 Here and below lowercase Roman indices may take values from 1 to N , where Nmay be larger than 3, but the summation convention is still supposed to apply.31.9 I admit that I am somewhat sloppy as regards the notation in (31.30) and onwardsin that Π is used as a functional of a vector v as well as a functional of the componentsvk of the vector–mathematicians would faint.31.10 In this book Potentials—see Chapter 32—are denoted Π, and the square of thelength falls in this class, which is the reason for using the somewhat strange notation.

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because in the sum over i all terms except the one for i = j vanish.

More generally, we may also be interested in the behavior of Scalar Prod-Scalar product

= Inner product ucts, also denoted Inner Products, of vectors or matrices. As an introductionconsider the scalar product of two vectors u and v with components ui andvj , respectively

Inner product of

vectors u and vΦ(ui, vj) = u · v = uivi (31.35)

As before we may wish to discover the behavior of the functional Φ closeto its value for uk and vm. Therefore, we differentiate Φ with respect tofixed components of u and v, namely uk and vm

∂Φ(ui, vj)

∂uk=∂ui∂uk

vi = δikvi = vk

∂Φ(ui, vj)

∂vm=

∂vi∂vm

ui = δimui = um

(31.36)

Finally, compute the derivatives of the product of two vectors ui and vjwith a two-dimensional matrix Aij whose elements do not depend on eitherof the vectors. The product Φ is defined as

Φ(ui, vj) = Aijuivj or Φ(u,v) = uTAv = vTATu (31.37)

where T signifies the matrix transpose. The partial derivatives then are

∂Φ(ui, vj)

∂uk= Aij

∂ui∂uk

vj = Aijδikvj = Akjvj = vTAT

∂Φ(ui, vj)

∂vm= Aij

∂vj∂vm

ui = Aijδjmui = Aimui = Au

(31.38)

In Part I we repeatedly encounter products like the one in (31.37) exceptthat it contains only one vector

Π(ui) =12Aijuiuj (31.39)

where the factor 12 is chosen for convenience and consistency with most of

our applications, and where the matrix is symmetric

Aij = Aji (31.40)

Then, the partial derivative is

∂Π(ui)

∂uk= 1

2Aij∂(uiuj)

∂uk= 1

2Aij∂ui∂uk

uj +12Aij

∂uj∂uk

ui

= 12Aijδikuj +

12Aijδjkui =

12Akjuj +

12Aikui

= Aikui

(31.41)

where the symmetry of Aij has been exploited in the last step of the deriva-tion. In the Part I and II we carry out manipulations like the ones abovevery frequently and do not continue our efforts here.

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31.4.2 Functions as Generalized Coordinates

The idea of generalized coordinates may be extended further in that we may Functions as

Generalized

Coordinates

choose functions instead of vectors as the basis. We do not intend to gointo any depth here and at this point limit ourselves to a specific example.Consider the following formula31.11

Φ(u, v) =

∫ 1

0

u(x)2v(x)dx (31.42)

where the notation Φ(u, v) indicates that the value of Φ depends on thefunctions u and v. Obviously, u(x) and v(x) play the same role here asdo ui and vj in the preceding examples. Again, in some instances it isnecessary to investigate the behavior of Φ for functions that are close to uand v. Therefore we compute

∂Φ(u, v)

∂u= 2

∫ 1

0

u(x)v(x)dx∂Φ(u, v)

∂v=

∫ 1

0

u(x)2dx (31.43)

We do not continue along this line here, but refer to Chapter 32, and asregards applications to Parts I and II.

31.11 Do not try to give an interpretation of (31.42)—I didn’t.

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