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Chapter 2

Vectors, Tensors, and CurvilinearCoordinates

2.1 Introduction

Many machine parts and structures have curved surfaces which dictatethe use of curvilinear coordinate systems. In particular, it is difficult to

formulate a general theory of shells without an adequate knowledge aboutthe coordinates of curved surfaces.Although a rectangular system may be inscribed conveniently in certain

bodies, deformation carries the straight lines and planes to curved lines andsurfaces; rectangular coordinates are deformed into arbitrary curvilinear co-ordinates. Consequently, we are unavoidably led to explore the differentialgeometry of curved lines and surfaces using curvilinear coordinates, if onlyin the deformed body.

In general, coordinates need not measure length directly, as Cartesian

coordinates. The curvilinear coordinates need only define the position ofpoints in space, but they must do so uniquely and continuously if theyare to serve our purpose in the mechanics of continuous media. A familiarexample is the spherical system of coordinates consisting of the radial dis-tance and the angles of latitude and longitude: Spherical coordinates havethe requisite continuity except along the polar axis; one is a length and twoare angles, dimensionless in radians.

2003 by CRC Press LLC

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Figure 2.1 Curvilinear coordinate lines and surfaces

2.2 Curvilinear Coordinates, Base Vectors, and Metric

TensorThe position of point A in Figure 2.1 is given by the vector

r = xii, (2.1)

where xi are rectangular coordinates and i are unit vectors as shown.Let i denote arbitrary curvilinear coordinates. We assume the existence

of equations which express the variables xi in terms of i and vice versa;

that is,

xi = xi(1, 2, 3), i = i(x1, x2, x3). (2.2), (2.3)

Also, we assume that these have derivatives of any order required in thesubsequent analysis.


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Figure 2.2 Network of coordinate curves and surfaces

Suppose that xi = xi(a1, a2, a3) are the rectangular coordinates of point

A in Figure 2.1. Then xi(1, a2, a3) are the parametric equations of a curve

through A; it is the 1 curve of Figure 2.1. Likewise, the 2 and 3 curvesthrough point A correspond to fixed values of the other two variables.The equations xi = xi(

1, 2, a3) are parametric equations of a surface, the3 surface shown shaded in Figure 2.1. Similarly, the 1 and 2 surfacescorrespond to 1 = a1 and 2 = a2. At each point of the space there is anetwork of curves and surfaces (see Figure 2.2) corresponding to the trans-

formation of equations (2.2) and (2.3).By means of (2.2) and (2.3), the position vector r can be expressed inalternative forms:

r = r(x1, x2, x3) = r(1, 2, 3). (2.4)

A differential change di is accompanied by a change dri tangent to thei line; a change in 1 only causes the increment dr1 illustrated in Fig-

ure 2.1. It follows that the vector

gi r

i=

xji

j (2.5)

is tangent to the i curve. The tangent vector gi is sometimes called a basevector.


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Figure 2.3 Tangent and normal base vectors

Let us define another triad of vectors gi

such that

gi gj ij . (2.6)

The vector gi is often called a reciprocal base vector. Since the vectors gi aretangent to the coordinate curves, equation (2.6) means that the vectors gi

are normal to the coordinate surfaces. This is illustrated in Figure 2.3.We will call the triad gi tangent base vectors and the triad g

i normal base

vectors. In general they are not unit vectors.The triad gi can be expressed as a linear combination of the triad gi,

and vice versa. To this end we define coefficients gij and gij such that

gi gijgj , gi gijgj. (2.7), (2.8)

From equations (2.6) to (2.8), it follows that

gij = gji = gi gj , gij = gji = gi gj , (2.9), (2.10)

gimgjm = ij . (2.11)

The linear equations (2.11) can be solved to express gij in terms of gij , asfollows:

gij =cofactor of element gij in matrix [gij ]

|gij | . (2.12)


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The first-order differential of r is dr = r,i di gi di, and the corre-

sponding differential length d is given by

d2 = dr dr = gij di dj . (2.13)

Equation (2.13) is fundamental in differential geometry, and the coefficientsgij play a paramount role; they are components of the metric tensor. Lit-erally, a component gii provides the measure of length (per unit of

i)along the i line; a component gij (i = j) determines the angle betweeni and j lines.

For practical purposes, we may employ unit vectors ei and ei, tangentto the i line and normal to the i surface, respectively:

ei =gi

gii, ei =

gigii

. (2.14a, b)

If the coordinate system is orthogonal, then

gii = 1gii

, gij = gij = 0 (i = j).

In the rectangular Cartesian system of coordinates,

gij = gij = ij ij .

In accordance with (2.5) and (2.9),

gij =xki

xkj

. (2.15)

By the rules for multiplying determinants

|gij | =

xij

xji

=

xij

2

.

Since this quantity plays a central role in the differential geometry, weemploy the symbol

g |gij | =xij

2

. (2.16a, b)


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Also, from (2.11), it follows by the rules for multiplying determinants andby using (2.16a) that

|gimgjm | = |gij ||gij | = |gij |g = |ij | = 1,

and therefore

|gij | = 1g

. (2.16c)

2.3 Products of Base Vectors

The vector product of gi and gj is normal to their plane but zero ifi equals j. In general, the vector product has the form

gi gj = M ijkgk, (2.17a)

where M is a positive scalar and ijk is the permutation symbol definedby (1.5) and (1.6). Conversely,

gi gj = N ijkgk, (2.17b)

where N is another positive scalar. To determine M and N from (2.17a)and (2.17b), we form the scalar (dot) products of these equations with

gmijm and gmijm , respectively, and recall (1.12c) to obtain

M = 16

ijkgk (gi gj), (2.18a)

N = 16

ijk gk (gi gj). (2.18b)

Now, the vectors gi of (2.18a) can be expressed in terms of the reciprocalvectors gi according to (2.7); then

M = 16

ijkgimgjngkl gl (gm gn). (2.19a)

It follows from (2.19a), (2.17b), and (1.9) that

M = 16

ijkmnlgimgjngklN = |gij |N. (2.19b)


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In like manner,N =

|gij

|M. (2.20)

Recall (2.5) and (1.6):

gi r,i =xki

k, i j = ijk k.

Then (2.18a) assumes the forms:

M = 16

ijklmnxl

i

xm

j

xn

k=

xi

j

.

By the notation (2.16a)

M =

g. (2.21a)

In accordance with (2.16a), (2.19b), and (2.21a),

N = 1g

. (2.21b)

With the notations of (2.21a, b), the equations (2.17a, b) take the forms:

gi gj =

g ijk gk, gi gj =

ijkggk. (2.22a, b)

It follows that

g ijk = gk (gi gj),

ijkg

= gk (gi gj). (2.23a, b)

Let us define

eijk g ijk , eijk 1g

ijk . (2.24a, b)

Then, according to (2.22a, b) and (2.23a, b)

eijk = gk (gi gj), eijk = gk (gi gj), (2.25a, b)

gi gj = eijk gk, gi gj = eijkgk. (2.26a, b)


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Figure 2.4 Contravariant and covariant components of vectors

2.4 Components of Vectors

Any vector can be expressed as a linear combination of the base vectors gior gi. Thus, the vector V has alternative forms (see Figure 2.4):

V= Vigi = Vigi. (2.27a, b)

The alternative components follow by means of (2.6) to (2.8)

Vi = gi V, Vi = gi V, (2.28a, b)

Vi = gijVj , Vi = gijVj . (2.29a, b)

The components Vi and Vi are called contravariant and covariant compo-nents, respectively. The full significance of these adjectives is discussed inSection 2.7.

In general, the base vectors (gi, gi) are not unit vectors. The vector V

can also be expressed with the aid of the unit base vectors (ei, ei) defined

by (2.14a, b). By (2.14a, b) and (2.27a, b), we have

V= Vi

gii ei = Vi

gii ei. (2.30a, b)

The coefficients (Vi

gii ) and (Vi

gii ) are known as the physical compo-


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Figure 2.5 Physical components

nents of the vector V:

PVi Vi

gii, PVi = Vi

gii. (2.31a, b)

Notice that the physical components PVi and PVi are in directions tan-

gent to the i line and normal to the i surface, respectively. The physicalcomponents (2.31a, b) represent the lengths of the sides of the parallel-ograms formed by the vectors (Vigi) and (Vi g

i) of Figure 2.4. In thethree-dimensional case, these components are the lengths of the edges ofthe parallelepipeds constructed on the vectors (Vigi) or (PV

iei) and (Vi gi)

or (PViei) (see Figure 2.5). Furthermore, by forming the scalar product of

the vector V of equation (2.27b) with ej, by considering (2.14a), and the

orthogonality condition (2.6), it follows that the term Vi/gii is equal tothe length of the orthogonal projection ofV on the tangent to the i line.In a similar way, by multiplying (2.27a) with ej and in view of (2.14b) and

(2.6), it can be shown that the term Vi/

gii is equal to the length of theorthogonal projection ofV on the normal to the i surface.

The products of vectors assume alternative forms in terms of the con-travariant and covariant components. For example,

VU = ViUi = ViUi = ViUjg

ij = ViUjgij , (2.32ad)

VU = eijkViUjgk = eijkViUjgk, (2.33a, b)

U (VW) = eijkUiVjWk = eijkUiVjWk, (2.34a, b)

U (VW) = (UiWi)(Vjgj) (UiVi)(Wjgj). (2.35)


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Figure 2.6 Elemental parallelepiped

Recall that the result of the vector product VU is a vector normal to

the plane formed by the vectors V and U. The magnitude of this vector isequal to the area of the parallelogram with sides V and U. From (2.33a)it is apparent that the vector product satisfies the anticommutative law:

VU = UV. (2.36)

Recall also that the value of the scalar triple product U (VW) repre-sents the volume of the parallelepiped whose edges are the vectors U, V

and W. It follows from (2.34a), the definitions (2.24a), and the propertiesof the permutation symbol ijk (1.7) and (1.8) that

U (VW) = V (WU) = W (UV) (2.37a, b)

and

U (VW) = U (WV), (2.38a)

= V (UW), (2.38b)

= W (VU). (2.38c)


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2.5 Surface and Volume Elements

An elemental volume dv bounded by the coordinate surfaces through thepoints (1, 2, 3) and (1 + d1, 2 + d2, 3 + d3) is shown in Figure 2.6.In the limit, the volume element approaches

dv = dr1 (dr2 dr3)

= r,1 (r,2 r,3) d1 d2 d3

= g1 (g2 g3) d1 d2 d3

=

g d1 d2 d3. (2.39)

The elemental parallelepiped of Figure 2.6 is bounded by the coordinatesurfaces with areas dsi and unit normals

ei =

gigii

.

The area ds1 isds1 = e

1 (g2 g3) d

2 d3.

By (2.22a) we have

ds1 = e1 g1

g d2 d3

=

g11

g d2 d3. (2.40a)

Likewise,

ds2 =

g22

g d3 d1, (2.40b)

ds3 =

g33

g d1 d2. (2.40c)

In subsequent developments (see, e.g., Sections 2.11 and 3.3), we enclosea region by an arbitrary surface s. Then, we require a relation whichexpresses an element ds of that surface in terms of the adjacent elements

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Figure 2.7 Elemental tetrahedron

dsi of the coordinate surfaces. To that end, we examine the tetrahedronof Figure 2.7 which is bounded by triangular elements of the coordinatesurfaces si and the triangular element of the inclined surface s. The areas

of these elements are (1/2)dsi and (1/2)ds, where the dsi correspond to theareas of the quadrilateral elements of Figure 2.6 and equations (2.40ac).In keeping with the preceding development, one can view the inclined faceas an element of another coordinate surface, viz., the 1 surface ( lines 2

and 3 lie in surface s s1 ). The edges of the inclined triangular face havelength

g22 d

2 and

g33 d3; these are first-order approximations (valid

in the limit: di, di 0). The corresponding area of the inclined surfaceis [see (2.40a)]

12

ds

12

ds1 =12g

11

g d2 d 3.

For such enclosed tetrahedron

12

dsi ei = 1

2ds n, (2.41)

wherein n is the unit normal to s as ei is the unit normal to the surfacesi. Our reference to a second coordinate system

i is merely an artifice toidentify the area ds1 with the previous formulas for the area dsi. Equa-tion (2.41) holds for any surface s with unit normal n and any regular coor-dinate system i at a point of the surface. Ifn = nig

i, then in accordancewith (2.40ac), equation (2.41) is expressed by three scalar equations:

g1 (dsi ei) =

g d2 d3 = ds n1, (2.42a)

g2 (dsi ei) =

g d3 d1 = ds n2, (2.42b)


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g3 (dsi ei) =

g d1 d2 = ds n3. (2.42c)

2.6 Derivatives of Vectors

In accordance with (2.5)

gi,j

gij

=2r

i j=

2xpi j

p. (2.43ac)

Since the derivatives are supposed to be continuous,

gi,j = gj,i. (2.44)

The vector gi,j can be expressed as a linear combination of the base vec-

tors gi or gi:

gi,j = ijkgk = kijgk, (2.45a, b)

where the coefficients are defined by

ijk gk gi,j , kij gk gi,j . (2.46a, b)

The coefficients ijk and kij are known as the Christoffel symbols of the

first and second kind, respectively. According to (2.44) and (2.46a, b), the

symbols are symmetric in two lower indices; that is,

ijk = jik , kij =

kji .

From (2.9), (2.44), and (2.46a) it follows that

ijk =12

(gik,j + gjk,i gij,k). (2.47)

Using (2.7), (2.8), and (2.46a, b), we obtain the following relations:

ijk = gkllij ,

kij = g

klijl . (2.48a, b)

Differentiating (2.6) and employing (2.45a), we obtain

gi,k gj = gi gj,k = ijk .


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It follows thatgi,k =

ijkg

j . (2.49)

The partial derivative of an arbitrary vector V (2.27a, b) has the alter-native forms

V,i = Vj,igj + V

jgj,i = Vj,igj + Vjg

j,i.

In accordance with (2.45b) and (2.49),

V,i = (Vj,i + V

kjki)gj Vj |i gj , (2.50a, b)

but also

V,i = (Vj,i Vkkji)gj Vj |i gj . (2.51a, b)

Equations (2.50b) and (2.51b) serve to define the covariant derivatives ofthe contravariant (Vi) and covariant (Vi) components of a vector.

Observe that the covariant derivative (Vj |i or Vj |i) plays the same roleas the partial derivative (Vj,i) plays in the Cartesian coordinate system,

that the base vector (gi or gi) plays the same role as the unit vector i inthe Cartesian system, and that the metric tensor (gij or g

ij) reduces to theKronecker delta ij in the Cartesian coordinates.

From the definition (2.46a), the Christoffel symbol ijk in one coordinatesystem i is expressed in terms of the symbols in another system i by theformula:

ijk = lmnl

im

jn

k+ glm

l

k2m

ij. (2.52)

Equation (2.52) shows that the Christoffel symbols are not components ofa tensor (see Section 2.7).

From the definitions (2.16a), from (2.9), (2.12), and (2.49), it follows that

g

gij= ggij

and1

g

g

i= jji . (2.53)


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2.7 Tensors and Invariance

In the preceding developments we have introduced indicial notations,which facilitate the analyses of tensors and invariance. Also, certain ter-minology has been introduced, e.g., covariant, contravariant and invariant.We have yet to define tensors, tensorial transformations, and to demon-strate their roles in our analyses, most specifically the invariance of thosequantities which are independent of coordinates. These specifics are setforth in the present section.

Recall that a system of nth order has n free indices and contains 3n

components in our three-dimensional space. If the components of a system,which are expressed with respect to a coordinate system i, are transformedto another coordinate system i according to certain transformation laws,then the system of nth order is termed a tensor of nth order. The impor-tant attributes of tensors (e.g., invariant properties) are a consequence ofthe transformations which define tensorial components. The explicit ex-pressions of tensorial transformations follow:

Consider a transformation from one coordinate system i

to anotheri

,that is to say,

i = i(1, 2, 3), i = i(1, 2, 3).

The differentials of the variables i transform as follows:

d

i

=

i

j d

j

.

This linear transformation is a prototype for the transformation of thecomponents of a contravariant tensor.

In general, Ti(1, 2, 3) and Ti(1, 2, 3) are components of a first-ordercontravariant tensor in their respective systems, if

Ti

=

i

j Tj

, Ti

=

i

j Tj

. (2.54a, b)

The component of a first-order tensor is distinguished by one free index anda contravariant tensor by the index appearing as a superscript.

The functions Fij(1, 2, 3) with n superscripts are the components ofan nth-order contravariant tensor, if the components Fij(1, 2, 3) are


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given by the transformation:

Fij

n superscripts

= i

pj

q

n partial derivatives

Fpq .

n superscripts

(2.55)

Consider the transformation of the tangent base vectors gi(i) to another

coordinate system gi(i); by the chain rule for partial derivatives, we have

gi ri

= rj

j

i

j

igj . (2.56)

This linear transformation is the prototype for the transformation of thecomponents of a covariant tensor.

The functions Pij(1, 2, 3) with n subscripts are components of an

nth-order covariant tensor, if the components Pij(1, 2, 3) are given by

the transformation:

Pij

n subscripts

=p

iq

j


Ppq .

n subscripts

(2.57)

A tensor may have mixed character, partly contravariant and partly co-variant. The order of contravariance is given by the number of superscriptsand the order of covariance by the number of subscripts. The components

of a mixed tensor of order (m + n), contravariant of order m and covariantof order n, transform as follows:

m superscripts

Tijkl

n subscripts

=

m partial derivatives i

pj

q

r

ks

l


m superscripts Tpqrs .

n subscripts

(2.58)

Observe that the transformations (2.55), (2.57), and (2.58) express thetensorial components in one system i as a linear combination of the com-ponents in another i. The distinction between the contravariance (super-scripts) and covariance (subscripts) is crucial from the mathematical andphysical viewpoints. First, we observe that addition of tensorial compo-nents is meaningful if, and only if, they are of the same form, the same


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order of contravariance and covariance; then the sum is also the compo-nent of a tensor of that same order: e.g., if Aijk and B

ijk are components

of tensors in a system i, then (Cijk = Aijk + Bijk ) is also the componentof a tensor in that system. The proof follows from (2.58). Second, theproduct of tensorial components is also the component of a tensor: e.g., ifTij and Smn are components of tensors in a system

i, then the productQijmn TijSmn is also the component of a tensor in that system. Note thatthe latter is a tensor of fourth order. Again, the proof follows directly fromthe transformation (2.58).

Consider a summation of the form Qijmi; this might be termed a contrac-

tion, wherein a system of second order is contracted from one of fourthorder Qijmn by the repetition of the index i and the implied summation.According to (2.58)

Qklpq =k

il

jm

pn

qQijmn.

The contracted system has the components

Qklpk =k

in

kl

jm

pQijmn.

Note thatk

in

k=

n

i= ni .

Therefore,

Qklpk = l

j

m

pQijmi.

In words, the latter components are the components of a second-order ten-sor; it is a mixed tensor of first-order contravariant (one free superscript)and first-order covariant (one free subscript). It is especially important toobserve that one repeated index is a superscript and one is a subscript;one indicating the contravariant character, the other indicating the co-variant character. Such repetition of indices and implied summation (one

superscriptone subscript) is an inviolate rule to retain the tensorial char-acter.

Let us now consider the further contraction Qijij = TijSij . Again, T

ij

and Sij , hence Qijij are tensorial components. In the light of (2.55) and

(2.57)

TijSij =i

mj

nTmn

p

iq

jSpq.


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According to the chain rule for partial differentiation

TijSij = p

mq

nTmnSpq =

pm

qnT

mnSpq = TmnSmn. (2.59)

In words, this quantity is unchanged by a coordinate transformation. Suchquantities are invariants; they have the same value independently of thecoordinate system. The invariance hinges on the notions of covariance andcontravariance. An invariant function of curvilinear coordinates is obtainedby summations involving repeated indices which appear once as a super-

script and once as a subscript. Invariants have special physical meaningbecause they are not dependent on the choice of coordinates. They are eas-ily recognized as zero-order tensors (no free indices). However, care mustbe taken that repeated indices appear once as a superscript and once asa subscript, for otherwise the sum is not invariant. Cartesian coordinatesare the exception because the covariant and contravariant transformationsare then identical. For example, the Kronecker delta ij is a tensor in theCartesian system xi. It can be written with indices up or down, that is,ij =

ij = ij . Note that gij and gij are the contravariant and covariant

components obtained by the appropriate transformations of ij from therectangular to the curvilinear coordinate system:

gij =xki

xlj

kl =xki

xkj

, gij =i

xk

j

xlkl =

i

xk

j

xk;

ij are components of the metric tensor in a Cartesian coordinate system.Recall the definition (2.6) of the reciprocal vector gi and also (2.56).

Thengi gj =

ij = g

i gk

k

j.

The latter holds generally if, and only if,

gi = gmi

m. (2.60)

It follows from (2.56) and (2.60) that the components gij and gij transformaccording to the rules for covariant and contravariant components, respec-tively. Similarly, from equations (2.25a, b), the components eijk and e

ijk

are, respectively, covariant and contravariant:

eijk =xli

xmj

xnk

lmn, eijk =

i

xl

j

xm

k

xnlmn.


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Note: The Christoffel symbols do not transform as components of atensor; see equation (2.52).

A mathematical theoremknown as the quotient lawproves to be use-ful in establishing the tensor character of a system without recourse to thetransformation [i.e., (2.55), (2.57), and (2.58)]: If the product of a systemSpr...ij... with an arbitrary tensor is itself a tensor, then S

pr...ij... is also a tensor

(for proof see, e.g., J. C. H. Gerretsen [6], Section 4.2.3, p. 46).

Note the significant features of tensors, the significance of the notations(superscripts signify contravariance, subscripts covariance), the summationconvention, the linear transformations and the identification ofinvariance.Invariants are especially important to describe physical attributes whichare independent of coordinates; the conventions enable one to establishsuch quantities. The linear transformation of tensorial components is alsovery useful: If all components vanish in one system, then all vanish in everyother system. This means that any equation in tensorial form holds in everycoordinate system.

2.8 Associated Tensors

Let Tij denote the component of a covariant tensor. The component ofan associated contravariant tensor is

Tpq = gpigqjTij . (2.61)

Because gij

and Tij are components of contravariant and covariant tensors,respectively, the reader can show that Tpq is the component of a contravari-ant tensor. Moreover,

Tij = gpigqjTpq. (2.62)

The components of the covariant, contravariant, or mixed associated tensorsof any order are formed by raising or lowering indices as in (2.61) and (2.62),that is, by multiplying and summing with the appropriate versions of themetric tensor, gij or gij . Observe that the base vectors conform to this rule

according to (2.7) and (2.8).If the tensors are not symmetric in two indices, it is essential that the

proper position of the indices is preserved when raising or lowering indices.For example, this can be accomplished by placing a dot ( ) in the vacantposition:

Tji = gipTpj, Tij = g

ipTpj ,


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Tij = gipTpj = gjpT

pi , T

ij = gjpTip = gipTjp .

If Tij is symmetric, that is, Tij = Tji , then there is no need to mark theposition:

Tji = Tji = T

ji .

2.9 Covariant DerivativeThe essential feature of the covariant derivative is its tensor character.

For example, Vj|i of (2.50b): Since Vj |i = gj V,i, it follows from thetensorial transformation (Section 2.7) that Vj |i transform as componentsof a mixed tensor.

Because of the invariance property, it is useful to define covariant differ-entiation for tensors of higher order. Equation (2.50b) is the prototype forthe covariant derivative of a contravariant tensor. The general form for the

covariant derivative of a contravariant tensor of order m follows:

Fijm|p Fijm,p + Fqjmiqp + Fiqmjqp + + Fijqmqp. (2.63)

The covariant derivative of (2.63) includes the partial derivative (under-lined) but augmented by m additional terms, each is a sum of productsin which a component Fqjm is multiplied by a Christoffel symbol iqp; ineach of the latter, the index p of the independent variable of differentia-

tion and the dummy index q appear as subscripts, and a different index ofthe component Fij is the superscript; that superscript on Fij, which isreplaced by the dummy index q.

Equation (2.51b) is the prototype for the covariant derivative of a covari-ant component. The general form for a tensor of order n follows:

Pijn|p Pijn,p Prjnrip Pirnrjp Pijrrnp. (2.64)

The covariant derivative again includes the partial derivative (underlined)but again it is augmented by n additional terms, each is a sum of productsin which a component Prjn is multiplied by a Christoffel symbol

rip; here

each symbol has the index p of the variable of differentiation as a subscript,the dummy index r as a superscript and successive subscripts of the com-ponent Pij appear as the second subscript on the Christoffel symbol

rip;

that subscript on Pij, which is replaced by the dummy index r.


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The covariant derivative of a mixed component follows the rules for thecontravariant and covariant components: The partial derivative is aug-

mented by terms formed according to (2.63) or (2.64) as the componenthas contravariant or covariant character, respectively:

Tijmkln |p = T

ijmkln,p + T

qjmkln

iqp + + T

ijqkln

mqp

Tijmrln rkp T

ijmklr

rnp. (2.65)

In general, the order of covariant differentiation is not permutable. IfAjdenotes a covariant component of a tensor, then generally

Aj |kl = Aj |lk.

According to (2.64)

Aj |kl Aj|lk = RijklAi, (2.66)

where

Rijkl ijl,k ijk,l + mjl imk mjk iml. (2.67)

These comprise the components of the Riemann-Christoffel tensor or theso-called mixed-curvature tensor. The significance of the latter term is ap-parent when we observe that all components vanish in a system of Cartesiancoordinates; then Aj |kl = Aj |lk = Aj,kl. The components in any coordi-nates of an Euclidean space can be obtained by a linear transformation,i.e., the appropriate tensorial transformation from Cartesian components.It follows that the Riemann-Christoffel tensor vanishes in Euclidean space.

This fact imposes geometrical constraints upon the deformation of contin-uous bodies, as described in the subsequent chapters.

The associated curvature tensor has the components

Rijkl = gimRm jkl, (2.68a)

= lji,k kji,l + mjk ilm

mjl ikm, (2.68b)

= 12 (gjk,il + gil,jk gik,jl gjl,ik)

+ gmn(jkm iln jlm ikn). (2.68c)

From the definition (2.68c) and the symmetries of the components of themetric tensor (gij = gji) and of the Christoffel symbol (ijk = jik) it fol-


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lows that

Rijkl = Rjikl, (2.69a)

Rijkl = Rijlk, (2.69b)

Rijkl = Rklij. (2.69c)

2.10 Transformation from Cartesian to CurvilinearCoordinates

According to (2.55), (2.57) and (2.58), if all components of a tensor van-ish in one coordinate system, then they vanish in every other. This meansthat an equation (or equations) expressed in tensorial form holds in everysystem. This is especially important in any treatment of a physical prob-lem, since any coordinate system is merely an artifice, which is introducedfor mathematical purposes: Often a Cartesian system is the simplest; anexpression in the Cartesian system may be a special form of a tensorialcomponent. Provided that the tensorial character is fully established, thegeneralization to another system is readily accomplished.

The Cartesian coordinates have several distinct features: The coordinateis length or, stated mathematically, the component of the metric tensor isthe Kronecker delta:

g

ij

= gij = ij

ij

i

j.

Also, the position of the suffix (superscript or subscript) is irrelevant. It fol-lows too that the Christoffel symbols vanish; hence, the covariant derivativereduces to the partial derivative, e.g.,

Vi|j = Vi|j = Vi,j = Vi,j .

Additionally, the components of vectors or second-order systems in orthog-onal directions have simpler interpretations, geometrically and physically.


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Figure 2.8 Region of integration

2.11 Integral Transformations

In the analysis of continuous bodies, especially by energy principles, in-tegrals arise in the form

I

v

AiB,i dv =

v

AiB,i

g d1 d2 d3, (2.70)

where the integration extends through a volume v. Ai and B are as-sumed continuous with continuous derivatives. The region of integration isbounded by the surface s in Figure 2.8. The entire bounding surface s is di-vided into surfaces s and s by the curve c such that 3 lines are tangent to salong c. If the surface s is irregular, e.g., possesses concave portions, thederivation must be amended; the surface s must be subjected to additionalsubdivision. Still the final result (2.72) holds.

One term of (2.70) is

I3

v

A3B,3

g d3 d1 d2. (2.71a)


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The integral (2.71a) can be rewritten as follows:

I3

s

s

A3B,3

g d3

d1 d2.

The term in braces can be integrated by parts so that

I3 =

A3B

g d1 d2

s

A3B

g d1 d2s

A3

g,3

B d1 d2 d3.

If ni represent the components of the unit normal vector n = nigi, then

according to (2.42c)g d1 d2

s

= n3 ds,

g d1 d2s

= n3 ds.

The negative sign in the second equation arises because n e3 = 1 on s.The integral I3 follows:

I3 =

s

A3B n3 ds

v

1g

A3

g,3

B dv. (2.71b)

The other terms of the integral I are similar. Consequently,

I

v

AiB,i dv =

s

AiB ni ds

v

1g

Ai

g,i

B dv. (2.72)

Equation (2.72) is one version of Greens theorem, one which is particularlyuseful in applications of the energy principles to continuous bodies. From(2.72) a series of useful formulations can be derived.

In a subsequent application of (2.72) we encounter vectors si, whichtransform as contravariant tensorial components, and a vector V. Then

from (2.72) it follows thatv

si V,i dv =

s

(si ni) V ds

v

1g

g si

,i

V dv.

(2.73)Integral transformations, which express certain surface integrals in terms

of line integrals are presented in Section 8.11.
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