Chapter 2 Vectors, Tensors, and Curvilinear Coordinates

2.1

Introduction

Many machine parts and structures have curved surfaces which dictate the use of curvilinear coordinate systems. In particular, it is difficult to formulate a general theory of shells without an adequate knowledge about the 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 and surfaces; rectangular coordinates are deformed into arbitrary curvilinear coordinates. Consequently, we are unavoidably led to explore the differential geometry of curved lines and surfaces using curvilinear coordinates, if only in the deformed body. In general, coordinates need not measure length directly, as Cartesian coordinates. The curvilinear coordinates need only define the position of points in space, but they must do so uniquely and continuously if they are to serve our purpose in the mechanics of continuous media. A familiar example is the spherical system of coordinates consisting of the radial distance and the angles of latitude and longitude: Spherical coordinates have the requisite continuity except along the polar axis; one is a length and two are angles, dimensionless in radians.

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Figure 2.1 Curvilinear coordinate lines and surfaces

2.2

Curvilinear Coordinates, Base Vectors, and Metric Tensor

The position of point A in Figure 2.1 is given by the vector r = xiˆıi ,

(2.1)

where xi are rectangular coordinates and ˆıi are unit vectors as shown. Let θi denote arbitrary curvilinear coordinates. We assume the existence of equations which express the variables xi in terms of θi and vice versa; that is, xi = xi (θ1 , θ2 , θ3 ),

θi = θi (x1 , x2 , x3 ).

(2.2), (2.3)

Also, we assume that these have derivatives of any order required in the subsequent analysis. © 2003 by CRC Press LLC

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 (θ1 , a2 , a3 ) are the parametric equations of a curve through A; it is the θ1 curve of Figure 2.1. Likewise, the θ2 and θ3 curves through point A correspond to fixed values of the other two variables. The equations xi = xi (θ1 , θ2 , a3 ) are parametric equations of a surface, the θ3 surface shown shaded in Figure 2.1. Similarly, the θ1 and θ2 surfaces correspond to θ1 = a1 and θ2 = a2 . At each point of the space there is a network of curves and surfaces (see Figure 2.2) corresponding to the transformation of equations (2.2) and (2.3). By means of (2.2) and (2.3), the position vector r can be expressed in alternative forms: r = r(x1 , x2 , x3 ) = r(θ1 , θ2 , θ3 ).

(2.4)

A differential change dθi is accompanied by a change dr i tangent to the θi line; a change in θ1 only causes the increment dr 1 illustrated in Figure 2.1. It follows that the vector gi ≡

∂r ∂xj ˆıj = ∂θi ∂θi

(2.5)

is tangent to the θi curve. The tangent vector g i is sometimes called a base vector. © 2003 by CRC Press LLC

Figure 2.3 Tangent and normal base vectors Let us define another triad of vectors g i such that g i · g j ≡ δ ij .

(2.6)

The vector g i is often called a reciprocal base vector. Since the vectors g i are tangent to the coordinate curves, equation (2.6) means that the vectors g i are normal to the coordinate surfaces. This is illustrated in Figure 2.3. We will call the triad g i tangent base vectors and the triad g i normal base vectors. In general they are not unit vectors. The triad g i can be expressed as a linear combination of the triad g i , and vice versa. To this end we define coefficients g ij and gij such that g i ≡ gij g j ,

g i ≡ g ij g j .

(2.7), (2.8)

From equations (2.6) to (2.8), it follows that gij = gji = g i · g j ,

g ij = g ji = g i · g j ,

g im gjm = δji .

(2.9), (2.10) (2.11)

The linear equations (2.11) can be solved to express g ij in terms of gij , as follows: cofactor of element gij in matrix [gij ] g ij = . (2.12) |gij | © 2003 by CRC Press LLC

The first-order differential of r is dr = r ,i dθi ≡ g i dθi , and the corresponding differential length d is given by d2 = dr · dr = gij dθi dθj .

(2.13)

Equation (2.13) is fundamental in differential geometry, and the coefficients gij play a paramount role; they are components of the metric tensor. Literally, a component gii provides the measure of length (per unit of θi ) along the θi line; a component gij (i = j) determines the angle between θi and θj lines. ˆi , tangent ˆi and e For practical purposes, we may employ unit vectors e i i to the θ line and normal to the θ surface, respectively: gi ˆi =
. e g ii

g ˆi =
i , e gii

(2.14a, b)

If the coordinate system is orthogonal, then g ii =

1 , gii

g ij = gij = 0

(i = j).

In the rectangular Cartesian system of coordinates, g ij = gij = δ ij ≡ δij . In accordance with (2.5) and (2.9), gij =

∂xk ∂xk . ∂θi ∂θj

(2.15)

By the rules for multiplying determinants       ∂xi   ∂xj   ∂xi 2      |gij | =  j   i  =  j  . ∂θ ∂θ ∂θ Since this quantity plays a central role in the differential geometry, we employ the symbol    ∂xi 2  g ≡ |gij | =  j  . ∂θ © 2003 by CRC Press LLC

(2.16a, b)

Also, from (2.11), it follows by the rules for multiplying determinants and by using (2.16a) that |g im gjm | = |g ij ||gij | = |g ij |g = |δji | = 1, and therefore |g ij | =

2.3

1 . g

(2.16c)

Products of Base Vectors

The vector product of g i and g j is normal to their plane but zero if i equals j. In general, the vector product has the form g i × g j = M ijk g k ,

(2.17a)

where M is a positive scalar and ijk is the permutation symbol defined by (1.5) and (1.6). Conversely, g i × g j = N ijk g k ,

(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 g m ijm and g m ijm , respectively, and recall (1.12c) to obtain M = 16 ijk g k · (g i × g j ),

(2.18a)

N = 16 ijk g k · (g i × g j ).

(2.18b)

Now, the vectors g i of (2.18a) can be expressed in terms of the reciprocal vectors g i according to (2.7); then M = 16 ijk gim gjn gkl g l · (g m × g n ).

(2.19a)

It follows from (2.19a), (2.17b), and (1.9) that M = 16 ijk mnl gim gjn gkl N = |gij |N. © 2003 by CRC Press LLC

(2.19b)

In like manner,

N = |g ij |M.

(2.20)

Recall (2.5) and (1.6): g i ≡ r ,i =

∂xk ˆık , ∂θi

ˆıi × ˆıj = ijk ˆık .

Then (2.18a) assumes the forms: M = 16 ijk lmn

   ∂xi  ∂xl ∂xm ∂xn   =  ∂θj  . ∂θi ∂θj ∂θk

By the notation (2.16a) M =

√

g.

(2.21a)

In accordance with (2.16a), (2.19b), and (2.21a), 1 N = √ . g

(2.21b)

With the notations of (2.21a, b), the equations (2.17a, b) take the forms: gi × gj =

√

g ijk g k ,

ijk gi × gj = √ gk . g

(2.22a, b)

ijk √ = g k · (g i × g j ). g

(2.23a, b)

It follows that √

g ijk = g k · (g i × g j ),

Let us define eijk ≡

√

g ijk ,

1 eijk ≡ √ ijk . g

(2.24a, b)

Then, according to (2.22a, b) and (2.23a, b) eijk = g k · (g i × g j ), g i × g j = eijk g k , © 2003 by CRC Press LLC

eijk = g k · (g i × g j ),

(2.25a, b)

g i × g j = eijk g k .

(2.26a, b)

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 g i or g i . Thus, the vector V has alternative forms (see Figure 2.4): V = V i g i = Vi g i .

(2.27a, b)

The alternative components follow by means of (2.6) to (2.8) V i = gi · V ,

Vi = g i · V ,

(2.28a, b)

V i = g ij Vj ,

Vi = gij V j .

(2.29a, b)

The components V i and Vi are called contravariant and covariant components, respectively. The full significance of these adjectives is discussed in Section 2.7. In general, the base vectors (g i , g i ) are not unit vectors. The vector V ˆi ) defined can also be expressed with the aid of the unit base vectors (ˆ ei , e by (2.14a, b). By (2.14a, b) and (2.27a, b), we have V =Vi The coefficients (V i






ˆ i = Vi gii e

gii ) and (Vi

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ˆi . g ii e

(2.30a, b)

g ii ) are known as the physical compo-

Figure 2.5 Physical components nents of the vector V : PV

i

≡Vi



 gii ,

P Vi

= Vi

g ii .

(2.31a, b)

Notice that the physical components P V i and P Vi are in directions tangent to the θi line and normal to the θi surface, respectively. The physical components (2.31a, b) represent the lengths of the sides of the parallelograms formed by the vectors (V i g i ) and (Vi g i ) of Figure 2.4. In the three-dimensional case, these components are the lengths of the edges of ˆi ) and (Vi g i ) the parallelepipeds constructed on the vectors (V i g i ) or (P V i e i ˆ ) (see Figure 2.5). Furthermore, by forming the scalar product of or (P Vi e ˆj , by considering (2.14a), the vector V of equation (2.27b) with e and the
orthogonality condition (2.6), it follows that the term Vi / gii is equal to the length of the orthogonal projection of V on the tangent to the θi line. ˆj and in view of (2.14b) and In a similar way, by multiplying (2.27a) with
e i (2.6), it can be shown that the term V / g ii is equal to the length of the orthogonal projection of V on the normal to the θi surface. The products of vectors assume alternative forms in terms of the contravariant and covariant components. For example, V · U = V i Ui = Vi U i = Vi Uj g ij = V i U j gij ,

(2.32a–d)

V × U = eijk V i U j g k = eijk Vi Uj g k ,

(2.33a, b)

U · (V × W ) = eijk U i V j W k = eijk Ui Vj Wk ,

(2.34a, b)

U × (V × W ) = (U i Wi )(V j g j ) − (U i Vi )(W j g j ). © 2003 by CRC Press LLC

(2.35)

Figure 2.6 Elemental parallelepiped Recall that the result of the vector product V × U is a vector normal to the plane formed by the vectors V and U . The magnitude of this vector is equal 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: V × U = −U × V .

(2.36)

Recall also that the value of the scalar triple product U · (V × W ) represents 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 properties of the permutation symbol ijk (1.7) and (1.8) that U · (V × W ) = V · (W × U ) = W · (U × V )

(2.37a, b)

and U · (V × W ) = −U · (W × V ),

(2.38a)

= −V · (U × W ),

(2.38b)

= −W · (V × U ).

(2.38c)

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2.5

Surface and Volume Elements

An elemental volume dv bounded by the coordinate surfaces through the points (θ1 , θ2 , θ3 ) and (θ1 + dθ1 , θ2 + dθ2 , θ3 + dθ3 ) is shown in Figure 2.6. In the limit, the volume element approaches dv = dr 1 · (dr 2 × dr 3 ) = r ,1 · (r ,2 × r ,3 ) dθ1 dθ2 dθ3 = g 1 · (g 2 × g 3 ) dθ1 dθ2 dθ3 =




g dθ1 dθ2 dθ3 .

(2.39)

The elemental parallelepiped of Figure 2.6 is bounded by the coordinate surfaces with areas dsi and unit normals gi ˆi =
. e g ii The area ds1 is

ˆ1 · (g 2 × g 3 ) dθ2 dθ3 . ds1 = e

By (2.22a) we have ˆ1 · g 1 ds1 = e




g dθ2 dθ3

=



g 11 g dθ2 dθ3 .

(2.40a)

ds2 =



g 22 g dθ3 dθ1 ,

(2.40b)

ds3 =



g 33 g dθ1 dθ2 .

(2.40c)

Likewise,

In subsequent developments (see, e.g., Sections 2.11 and 3.3), we enclose a region by an arbitrary surface s. Then, we require a relation which expresses an element ds of that surface in terms of the adjacent elements © 2003 by CRC Press LLC

Figure 2.7 Elemental tetrahedron dsi of the coordinate surfaces. To that end, we examine the tetrahedron of Figure 2.7 which is bounded by triangular elements of the coordinate surfaces 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 the areas of the quadrilateral elements of Figure 2.6 and equations (2.40a–c). In keeping with the preceding development, one can view the inclined face as an element of another coordinate surface, viz., the θ¯1 surface ( lines θ¯2 and θ¯3
lie in surface s
≡ s¯1 ). The edges of the inclined triangular face have length g¯22 dθ¯ 2 and g¯33 dθ¯ 3 ; these are first-order approximations (valid in the limit: dθi , dθ¯i → 0). The corresponding area of the inclined surface is [see (2.40a)]

1 1 s1 = 12 g¯11 g¯ dθ¯ 2 dθ¯ 3 . 2 ds ≡ 2 d¯ For such enclosed tetrahedron 1 2

ˆi = dsi e

1 2

ˆ ds n,

(2.41)

ˆ is the unit normal to s as e ˆi is the unit normal to the surface wherein n si . Our reference to a second coordinate system θ¯i is merely an artifice to identify the area d¯ s1 with the previous formulas for the area dsi . Equaˆ and any regular coortion (2.41) holds for any surface s with unit normal n ˆ = ni g i , then in accordance dinate system θi at a point of the surface. If n with (2.40a–c), equation (2.41) is expressed by three scalar equations: ˆi ) = g 1 · (dsi e


g dθ2 dθ3 = ds n1 ,

(2.42a)

ˆi ) = g 2 · (dsi e


g dθ3 dθ1 = ds n2 ,

(2.42b)

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ˆi ) = g 3 · (dsi e

2.6




g dθ1 dθ2 = ds n3 .

(2.42c)

Derivatives of Vectors

In accordance with (2.5) g i,j ≡

∂g i ∂2r ∂ 2 xp ˆıp . = = ∂θj ∂θi ∂θj ∂θi ∂θj

(2.43a–c)

Since the derivatives are supposed to be continuous, g i,j = g j,i .

(2.44)

The vector g i,j can be expressed as a linear combination of the base vectors g i or g i : g i,j = Γijk g k = Γkij g k ,

(2.45a, b)

where the coefficients are defined by Γijk ≡ g k · g i,j ,

Γkij ≡ g k · g i,j .

(2.46a, b)

The coefficients Γijk and Γkij 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, Γijk = Γjik ,

Γkij = Γkji .

From (2.9), (2.44), and (2.46a) it follows that Γijk = 12 (gik,j + gjk,i − gij,k ).

(2.47)

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

Γkij = g kl Γijl .

Differentiating (2.6) and employing (2.45a), we obtain g i,k · g j = −g i · g j,k = −Γijk . © 2003 by CRC Press LLC

(2.48a, b)

It follows that

g i,k = −Γijk g j .

(2.49)

The partial derivative of an arbitrary vector V (2.27a, b) has the alternative forms V,i = V j,i g j + V j g j,i = Vj,i g j + Vj g j ,i . In accordance with (2.45b) and (2.49), V,i = (V j ,i + V k Γjki )g j ≡ V j |i g j ,

(2.50a, b)

V,i = (Vj,i − V k Γkji )g j ≡ V j |i g j .

(2.51a, b)

but also

Equations (2.50b) and (2.51b) serve to define the covariant derivatives of the contravariant (V i ) and covariant (Vi ) components of a vector. Observe that the covariant derivative (V j |i or Vj |i ) plays the same role as the partial derivative (Vj,i ) plays in the Cartesian coordinate system, that the base vector (g i or g i ) plays the same role as the unit vector ˆıi in the Cartesian system, and that the metric tensor (gij or g ij ) reduces to the Kronecker delta δij in the Cartesian coordinates. From the definition (2.46a), the Christoffel symbol Γijk in one coordinate system θ¯i is expressed in terms of the symbols in another system θi by the formula: Γijk = Γlmn

∂θl ∂θm ∂θn ∂θl ∂ 2 θm + glm . i j k ¯ ¯ ¯ ∂θ ∂θ ∂θ ∂ θ¯k ∂ θ¯i ∂ θ¯j

(2.52)

Equation (2.52) shows that the Christoffel symbols are not components of a tensor (see Section 2.7). From the definitions (2.16a), from (2.9), (2.12), and (2.49), it follows that ∂g = gg ij ∂gij and

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1 ∂ g
= Γjji . g ∂θi

(2.53)

2.7

Tensors and Invariance

In the preceding developments we have introduced indicial notations, which facilitate the analyses of tensors and invariance. Also, certain terminology has been introduced, e.g., covariant, contravariant and invariant. We have yet to define tensors, tensorial transformations, and to demonstrate their roles in our analyses, most specifically the invariance of those quantities which are independent of coordinates. These specifics are set forth 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 θi , are transformed to another coordinate system θ¯i according to certain transformation laws, then the system of nth order is termed a tensor of nth order. The important attributes of tensors (e.g., invariant properties) are a consequence of the transformations which define tensorial components. The explicit expressions of tensorial transformations follow: Consider a transformation from one coordinate system θi to another θ¯i , that is to say, θ¯i = θ¯i (θ1 , θ2 , θ3 ),

θi = θi (θ¯1 , θ¯2 , θ¯3 ).

The differentials of the variables θ¯i transform as follows: dθ¯i =

∂ θ¯i j dθ . ∂θj

This linear transformation is a prototype for the transformation of the components of a contravariant tensor. In general, T i (θ1 , θ2 , θ3 ) and T i (θ¯1 , θ¯2 , θ¯3 ) are components of a first-order contravariant tensor in their respective systems, if Ti =

∂ θ¯i j T , ∂θj

Ti =

∂θi j T . ∂ θ¯j

(2.54a, b)

The component of a first-order tensor is distinguished by one free index and a contravariant tensor by the index appearing as a superscript. The functions F ij··· (θ1 , θ2 , θ3 ) with n superscripts are the components of an nth-order contravariant tensor, if the components F ij··· (θ¯1 , θ¯2 , θ¯3 ) are © 2003 by CRC Press LLC

given by the transformation: F ij··· 



= 

n superscripts

∂ θ¯i ∂ θ¯j ··· F pq··· . ∂θp ∂θq      

(2.55)

n partial derivatives n superscripts

Consider the transformation of the tangent base vectors g i (θi ) to another ¯ i (θ¯i ); by the chain rule for partial derivatives, we have coordinate system g gi ≡

∂r ∂r ∂ θ¯j ∂ θ¯j ¯j . = ≡ g ∂θi ∂ θ¯j ∂θi ∂θi

(2.56)

This linear transformation is the prototype for the transformation of the components of a covariant tensor. The functions P ij··· (θ1 , θ2 , θ3 ) with n subscripts are components of an nth-order covariant tensor, if the components P ij··· (θ¯1 , θ¯2 , θ¯3 ) are given by the transformation: P ij··· 



= 

n subscripts

∂θp ∂θq ··· P pq··· . ∂ θ¯i ∂ θ¯j      n partial derivatives

(2.57)

n subscripts

A tensor may have mixed character, partly contravariant and partly covariant. The order of contravariance is given by the number of superscripts and the order of covariance by the number of subscripts. The components of a mixed tensor of order (m + n), contravariant of order m and covariant of order n, transform as follows: m partial derivatives

m superscripts





T ij··· kl··· 







n subscripts

= 

  ∂ θ¯i ∂ θ¯j ··· ∂θp ∂θq

m superscripts

   ∂θ ∂θ pq··· Trs··· . ··· ∂ θ¯k ∂ θ¯l       r

s

n partial derivatives

(2.58)

n subscripts

Observe that the transformations (2.55), (2.57), and (2.58) express the tensorial components in one system θ¯i as a linear combination of the components in another θi . The distinction between the contravariance (superscripts) and covariance (subscripts) is crucial from the mathematical and physical viewpoints. First, we observe that addition of tensorial components is meaningful if, and only if, they are of the same form, the same © 2003 by CRC Press LLC

order of contravariance and covariance; then the sum is also the compoij nent of a tensor of that same order: e.g., if Aij k and Bk are components ij ij ij of tensors in a system θi , then (Ck = Ak + Bk ) is also the component of a tensor in that system. The proof follows from (2.58). Second, the product of tensorial components is also the component of a tensor: e.g., if T ij and Smn are components of tensors in a system θi , then the product ij Qij mn ≡ T Smn is also the component of a tensor in that system. Note that the latter is a tensor of fourth order. Again, the proof follows directly from the transformation (2.58). Consider a summation of the form Qij mi ; this might be termed a “contraction,” wherein a system of second order is contracted from one of fourth order Qij mn by the repetition of the index i and the implied summation. According to (2.58) Qkl pq =

∂ θ¯k ∂ θ¯l ∂θm ∂θn ij Q . ∂θi ∂θj ∂ θ¯p ∂ θ¯q mn

The “contracted” system has the components Qkl pk = Note that

∂ θ¯k ∂θn ∂ θ¯l ∂θm ij Q . ∂θi ∂ θ¯k ∂θj ∂ θ¯p mn

∂θn ∂ θ¯k ∂θn = = δin . ∂θi ∂ θ¯k ∂θi

Therefore, Qkl pk =

∂ θ¯l ∂θm ij Q . ∂θj ∂ θ¯p mi

In words, the latter components are the components of a second-order tensor; it is a mixed tensor of first-order contravariant (one free superscript) and first-order covariant (one free subscript). It is especially important to observe that one repeated index is a superscript and one is a subscript; one indicating the contravariant character, the other indicating the covariant character. Such repetition of indices and implied summation (one superscript–one subscript) is an inviolate rule to retain the tensorial character. ij ij Let us now consider the further “contraction” Qij ij = T Sij . Again, T ij and Sij , hence Qij are tensorial components. In the light of (2.55) and (2.57) ∂ θ¯i ∂ θ¯j mn ∂θp ∂θq T ij S ij = T Spq . ∂θm ∂θn ∂ θ¯i ∂ θ¯j © 2003 by CRC Press LLC

According to the chain rule for partial differentiation T ij S ij =

∂θp ∂θq mn p q mn T Spq = δm δn T Spq = T mn Smn . m n ∂θ ∂θ

(2.59)

In words, this quantity is unchanged by a coordinate transformation. Such quantities are invariants; they have the same value independently of the coordinate system. The invariance hinges on the notions of covariance and contravariance. An invariant function of curvilinear coordinates is obtained by summations involving repeated indices which appear once as a superscript and once as a subscript. Invariants have special physical meaning because they are not dependent on the choice of coordinates. They are easily recognized as zero-order tensors (no free indices). However, care must be taken that repeated indices appear once as a superscript and once as a subscript, for otherwise the sum is not invariant. Cartesian coordinates are the exception because the covariant and contravariant transformations are then identical. For example, the Kronecker delta δji is a tensor in the Cartesian system xi . It can be written with indices up or down, that is, δji = δ ij = δij . Note that g ij and gij are the contravariant and covariant components obtained by the appropriate transformations of δ ij from the rectangular to the curvilinear coordinate system: gij =

∂xk ∂xl ∂xk ∂xk δkl = , i j ∂θ ∂θ ∂θi ∂θj

g ij =

∂θi ∂θj kl ∂θi ∂θj δ = ; ∂xk ∂xl ∂xk ∂xk

δ ij are components of the metric tensor in a Cartesian coordinate system. Recall the definition (2.6) of the reciprocal vector g i and also (2.56). Then ∂θk ¯ j = δji = g ¯i · gk ¯i · g . g ∂ θ¯j The latter holds generally if, and only if, ¯i = gm g

∂ θ¯i . ∂θm

(2.60)

It follows from (2.56) and (2.60) that the components gij and g ij transform according to the rules for covariant and contravariant components, respectively. Similarly, from equations (2.25a, b), the components eijk and eijk are, respectively, covariant and contravariant: eijk =

∂xl ∂xm ∂xn lmn , ∂θi ∂θj ∂θk

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eijk =

∂θi ∂θj ∂θk lmn  . ∂xl ∂xm ∂xn

Note: The Christoffel symbols do not transform as components of a tensor; see equation (2.52). A mathematical theorem—known as the quotient law —proves to be useful in establishing the tensor character of a system without recourse to the transformation [i.e., (2.55), (2.57), and (2.58)]: If the product of a system pr... S pr... ij... with an arbitrary tensor is itself a tensor, then S 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 summation convention, the linear transformations and the identification of invariance. Invariants are especially important to describe physical attributes which are independent of coordinates; the conventions enable one to establish such quantities. The linear transformation of tensorial components is also very useful: If all components vanish in one system, then all vanish in every other system. This means that any equation in tensorial form holds in every coordinate system.

2.8

Associated Tensors

Let Tij denote the component of a covariant tensor. The component of an associated contravariant tensor is T pq = g pi g qj Tij .

(2.61)

Because g ij and Tij are components of contravariant and covariant tensors, respectively, the reader can show that T pq is the component of a contravariant tensor. Moreover, (2.62) Tij = gpi gqj T pq . The components of the covariant, contravariant, or mixed associated tensors of 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 the metric tensor, g ij 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 vacant position: T i·j = gip T pj , © 2003 by CRC Press LLC

T i·j = g ip Tpj ,

Tij = gip T p·j = gjp T i·p ,

T ij = g jp T i·p = g ip T p· j .

If T ij is symmetric, that is, T ij = T ji , then there is no need to mark the position: T i·j = T j·i = T ji .

2.9

Covariant Derivative

The essential feature of the covariant derivative is its tensor character. For example, V j |i of (2.50b): Since V j |i = g j · V,i , it follows from the tensorial transformation (Section 2.7) that V j |i transform as components of a mixed tensor. Because of the invariance property, it is useful to define covariant differentiation for tensors of higher order. Equation (2.50b) is the prototype for the covariant derivative of a contravariant tensor. The general form for the covariant derivative of a contravariant tensor of order m follows: F ij···m |p ≡ F ij···m,p + F qj···m Γiqp + F iq···m Γjqp + · · · + F ij···q Γm qp .

(2.63)

The covariant derivative of (2.63) includes the partial derivative (underlined) but augmented by m additional terms, each is a sum of products in which a component F qj···m is multiplied by a Christoffel symbol Γiqp ; in each of the latter, the index p of the independent variable of differentiation and the dummy index q appear as subscripts, and a different index of the component F ij··· is the superscript; that superscript on F ij··· , which is replaced by the dummy index q. Equation (2.51b) is the prototype for the covariant derivative of a covariant component. The general form for a tensor of order n follows: P ij···n |p ≡ P ij···n,p − P rj···n Γrip − P ir···n Γrjp · · · − P ij···r Γrnp .

(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 products in which a component Prj···n is multiplied by a Christoffel symbol Γrip ; 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 component Pij··· appear as the second subscript on the Christoffel symbol Γrip ; that subscript on Pij··· , which is replaced by the dummy index r. © 2003 by CRC Press LLC

The covariant derivative of a mixed component follows the rules for the contravariant and covariant components: The partial derivative is augmented by terms formed according to (2.63) or (2.64) as the component has contravariant or covariant character, respectively: ij···m ij···m ij···q m i T kl···n |p = T kl···n,p + T qj···m kl···n Γqp + · · · + T kl···n Γqp ij···m r ij···m r − T rl···n Γkp − · · · − T kl···r Γnp .

(2.65)

In general, the order of covariant differentiation is not permutable. If Aj denotes a covariant component of a tensor, then generally Aj |kl
= Aj |lk . According to (2.64)

where

Aj |kl − Aj |lk = Ri·jkl Ai ,

(2.66)

i m i Ri·jkl ≡ Γijl,k − Γijk,l + Γm jl Γmk − Γjk Γml .

(2.67)

These comprise the components of the Riemann-Christoffel tensor or the so-called mixed-curvature tensor . The significance of the latter term is apparent when we observe that all components vanish in a system of Cartesian coordinates; then Aj |kl = Aj |lk = Aj,kl . The components in any coordinates 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 continuous bodies, as described in the subsequent chapters. The associated curvature tensor has the components Rijkl = g im Rm· jkl , m = Γlji,k − Γkji,l + Γm jk Γilm − Γjl Γikm ,

(2.68a) (2.68b)

= 12 (gjk,il + gil,jk − gik,jl − gjl,ik ) + g mn (Γjkm Γiln − Γjlm Γikn ).

(2.68c)

From the definition (2.68c) and the symmetries of the components of the metric tensor (gij = gji ) and of the Christoffel symbol (Γijk = Γjik ) it fol© 2003 by CRC Press LLC

lows that

2.10

Rijkl = −Rjikl ,

(2.69a)

Rijkl = −Rijlk ,

(2.69b)

Rijkl = Rklij .

(2.69c)

Transformation from Cartesian to Curvilinear Coordinates

According to (2.55), (2.57) and (2.58), if all components of a tensor vanish in one coordinate system, then they vanish in every other. This means that an equation (or equations) expressed in tensorial form holds in every system. This is especially important in any treatment of a physical problem, since any coordinate system is merely an artifice, which is introduced for mathematical purposes: Often a Cartesian system is the simplest; an expression in the Cartesian system may be a special form of a tensorial component. Provided that the tensorial character is fully established, the generalization to another system is readily accomplished. The Cartesian coordinates have several distinct features: The coordinate is length or, stated mathematically, the component of the metric tensor is the Kronecker delta: g ij = gij = δij ≡ δ ij ≡ δji . Also, the position of the suffix (superscript or subscript) is irrelevant. It follows too that the Christoffel symbols vanish; hence, the covariant derivative reduces to the partial derivative, e.g., V i |j = V i |j = V i,j = Vi,j . Additionally, the components of vectors or second-order systems in orthogonal directions have simpler interpretations, geometrically and physically.

© 2003 by CRC Press LLC

Figure 2.8 Region of integration

2.11

Integral Transformations

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


I≡

v

Ai B,i dv =




v

Ai B,i



g dθ1 dθ2 dθ3 ,

(2.70)

where the integration extends through a volume v. Ai and B are assumed continuous with continuous derivatives. The region of integration is bounded by the surface s in Figure 2.8. The entire bounding surface s is divided into surfaces s¯ and s¯ by the curve c such that θ3 lines are tangent to s along c. If the surface s is irregular, e.g., possesses concave portions, the derivation must be amended; the surface s must be subjected to additional subdivision. Still the final result (2.72) holds. One term of (2.70) is


I3 ≡ © 2003 by CRC Press LLC

v

A3 B,3



g dθ3 dθ1 dθ2 .

(2.71a)

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


s¯  3 3 I3 ≡ dθ1 dθ2 . A B,3 g dθ s¯

The term in braces can be integrated by parts so that

   3      I3 = A B g dθ1 dθ2 s¯ − A3 B g dθ1 dθ2 s¯


−



A3



g ,3 B dθ1 dθ2 dθ3 .

ˆ = ni g i , then If ni represent the components of the unit normal vector n according to (2.42c)   g dθ1 dθ2 = n3 ds¯, g dθ1 dθ2 = −n3 d¯ s. s¯

s¯

ˆ ·e ˆ3 = −1 on s¯. The negative sign in the second equation arises because n The integral I3 follows:




1 

3  A3 g ,3 B dv. I3 = A B n3 ds − (2.71b) g s v The other terms of the integral I are similar. Consequently,







1 

i i  Ai g ,i B dv. (2.72) I≡ A B,i dv = A B ni ds − g v s v Equation (2.72) is one version of Green’s theorem, one which is particularly useful 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 , which transform as contravariant tensorial components, and a vector δV . Then from (2.72) it follows that







1  i

i i  s · δV,i dv = (s ni ) · δV ds − g s ,i · δV dv. g v s v (2.73) Integral transformations, which express certain surface integrals in terms of line integrals are presented in Section 8.11. © 2003 by CRC Press LLC