Chapter 1 - Cartesian Tensors

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1

(1.1)

Chapter1

Cartesian Tensors

1.1 Vectors

In order to specify completely certain physical quantities, such as temperature, energy

and mass, it is necessary to give only a real number. These quantities are referred to as

scalars. In order to specify completely certain other physical quantities, such as force,

moment, velocity and acceleration, it is necessary to give both their magnitude (a non-

negative number), their direction and their sense. These quantities are referred to as

vectors. They may be represented in a three-dimensional, Euclidian space by directed line

segments (arrows) whose length is proportional to the magnitude of the vectors and whose

direction and sense are those of the vectors.Two vectors are equal if they have the same magnitude, direction and sense.

One way of denoting vector quantities is by boldface Latin letters, i.e., a, b, c, whereas

their magnitude is represented as *a*,*b* and *c*, respectively.

The sum of two vectors a and b, as, for example, of two forces or of two velocities,

is a vector c, which is specified (as shown in Fig. 1.1) by the diagonal of the parallelogram

having as adjacent sides the vectors a and b. This rule of addition of two vectors is

referred to as the parallelogram rule.

The negative of a vector a is a vector

a having the magnitude and direction of a andreverse sense.

Any vector whose magnitude is equal to unity is referred to as a unit vector . Every

nvector a may be expressed as the product of its magnitude and the unit vector i having

the same direction and sense as the vector a.

Figure 1.1 Addition of two vectors.



Cartesian Tensors2

(1.2)

(1.3)

1.1.1 Components and Representation of a Vector

Every vector a can be expressed as a linear combination of three arbitrary noncoplanar

linearly independent vectors b, c, and d, referred to as base vectors. That is,

†

where m, p and q are real numbers. The vectors mb, pc and qd are referred to as the

components of the vector a with respect to the base vectors b, c and d. It can be shown

that the components of a vector with respect to a set of base vectors b, c and d are unique.

That is, for any vector a there exists only one set of real numbers m, p and q satisfying

1 2 3relation (1.2). As base vectors, we choose the three orthogonal unit vectors i , i and i

which lie along the positive directions of the set of right-handed rectangular system of ††

1 2 3axes (cartesian system of axes) x , x and x , respectively (see Fig. 1.2). Thus, we may

represent a vector a as a linear combination of the three right-handed orthogonal unit

1 2 3vectors i , i and i as

1 1 2 2 3 3The vectors a i , a i and a i (see Fig. 1.2) are the cartesian components of the vector a.

iHowever, in the sequel we will refer to the quantities a (i = 1, 2, 3) as the cartesian

components of the vector a. It is apparent that a vector may be specified by its three

Figure 1.2 Components of a vector. Figure 1.3 Direct ion cosines of a unit

vector.

1 2 3, n i i† In general a set of n vectors a , a , a ..., a is said to be linearly independent if the relation a m = 0

iis satisfied only when m = 0 (i = 1,2, ..., n). In the contrary case, the set of vectors is said to be linearly

dependent .

1 2 3 3†† We say that a set of axes x , x , x is right-handed, when a right hand screw placed parallel to the x axis

3 1 2moves in the direction of increasing x when turned from the x to the x axis.



Vectors 3

(1.4)

(1.5)

(1.6)

(1.7)

(1.8)

(1.9a)

cartesian components. Thus, if each one of the cartesian components of a vector with

respect to a rectangular system of axes is equal to the corresponding component of

another vector referred to the same rectangular system of axes, the two vectors are equal;

that is, their components referred to any rectangular system of axes are equal.Referring to Fig. 1.2, from geometric considerations, it can be seen that the magnitude

of the vector a is given by

nConsider a unit vector i

Referring to Fig. 1.3, we may conclude that

nn1 n2 n3where cos , cos and cos are called the direction cosines of the unit vector i or

of line . From geometric considerations we find that the direction cosines of a vector

are related by the following relation:

The sum of two or more vectors is the vector whose cartesian components are the sum

1 1of the corresponding cartesian components of the added vectors. For instance, if a = a i

2 2 3 3 1 1 2 2 3 3 1 1 2 2 3 3+ a i + a i and b = b i + b i + b i , then their sum is a vector c = c i + c i + c i whose

components are given as

A vector may be represented as follows:

1. By the symbolic representation employed heretofore, i.e., a, b, A, B, which does not

require a choice of a coordinate system.

12. By its three cartesian components with respect to a set of orthogonal unit vectors i ,

2 3i , and i , using one of the following notations:

(a) Indicial notation, as,



Cartesian Tensors4

(1.9b)

(1.10)

(1.11)

(1.12a)

(1.12b)

(1.12c)

(1.14)

(b) Matrix notation, as,

1.1.2 Scalar or Dot Product of Two Vectors

Let us consider two vectors a and b and let us denote the angle between them by

(0 ). The scalar or dot product of the two vectors is denoted by a b and is equal

to a scalar c whose magnitude is given by the following relation:

Notice that if a b = 0 and a 0, b 0, then the vectors a and b are mutually

i perpendicular. On the basis of definition (1.10), it is apparent that the unit vectors i

(i = 1, 2, 3) satisfy the following relations:

Moreover, it can be shown that

Using equations (1.11) and (1.12b), the scalar product of two vectors may be written in

terms of their cartesian components as

(1.13)

Notice that the scalar product of two vectors can be found by matrix multiplication. That

is,

1.1.3 Vector or Cross Product of Two Vectors

Consider two vectors a and b and denote by ( 0 ) the angle between them



Vectors 5

(1.15)

Figure 1.4 Vector product.

(see Fig. 1.4). The vector product or the cross product of the two vectors a and b is a

vector c (see Fig. 1.4) defined by the following relation:

nwhere i is the unit vector normal to the plane of the vectors a, b in the sense in which the

right-hand screw will move, if it is turned from a to b.

Referring to Fig. 1.4, the magnitude of the vector c is equal to the area of the

parallelogram OABCO. On the basis of definition (1.15), it is apparent that

a x b = b x a (1.16)

iMoreover, the right-handed orthogonal system of unit vectors i (i = 1, 2, 3) (see Fig. 1.2)satisfy the following relations:

1 1 2 2 3 3i x i = 0 i x i = 0 i x i = 0

1 2 3 2 3 1 3 1 2i x i = i i x i = i i x i = i (1.17)

2 1 3 3 2 1 1 3 2i x i = i i x i = i i x i = i

Furthermore, it can be shown that

a x (b + c) = a x b + a x c (1.18a)

(m a) x b = m (a x b) = a x (m b) (1.18b)

where m is a real number.

By direct multiplication, using relations (1.17) and (1.18a), the vector product of two

vectors a and b may be expressed as follows in terms of their components, referred to the

same right-handed rectangular system of axes

2 3 3 2 1 3 1 1 3 2 1 2 2 1 3a x b = (a b a b )i + (a b a b )i + (a b a b )i (1.19)

i i where a and b (i = 1, 2, 3) are the component of the vectors a and b in the directions of

1 2 3the unit vectors i , i , i .



Cartesian Tensors6

(1.20)

(1.21)

(1.22)

(1.23)

Relation (1.19) may be written in the following easy to remember determinant form:

When the components of two vectors a and b are referred to a left-handed rectangular

system of axes, a minus sign must be prefixed on the right-hand side of relations (1.17),

(1.19) and (1.20). In order to eliminate this difficulty, in this text we use only right-

handed rectangular systems of axes unless stated otherwise.

1.1.4 Rotation of a Rectangular System of Axes — Transformation Matrix

i jLet us consider two right-handed rectangular systems of axes x (i = 1, 2, 3) and x ( j

= 1, 2, 3) having the same origin at an arbitrary point O. Referring to Fig. 1.5, the

i jdirection cosines of the system of axes x with respect to the system of axes x are defined

as

or in indicial notation.

i j i jDenoting by i(i = 1, 2, 3) and i ( j = 1, 2, 3), the unit vectors acting along the x and x

axes, respectively, and referring to Fig. 1.5, we have

or

Similarly, we obtain

The nine direction cosines may be written in a matrix form as



Vectors 7

(1.24a)

(1.24b)

(1.25b)

(1.25a)

Figure 1.5 Rotation of a right-handed rectangular system of axes.

or

s iThe 3 x 3 matrix [ ] is referred to as the transformation matrix for the system of axes x

j(i = 1, 2, 3) with respect to the system of axes x ( j = 1, 2, 3). The nine direction cosines

ij i (i, j = 1, 2, 3) specify the rectangular system of axes x relative to the rectangular

jsystem of axes x but are not independent. They satisfy relations resulting from the

orthogonality of the two systems of axes. These relations are:

The above relations may be rewritten in the following form:



Cartesian Tensors8

(1.26a)

(1.26b)

(1.27)

(1.28)

(1.29)

(1.30)

(1.31)

(1.32)

Referring to relations (1.24), relations (1.25) may be written in matrix form as

where [ I ] is the 3 x 3 unit matrix defined by

Relations (1.26) are referred to as the conditions of orthog onality. Linear

transformations, such as (1.22) and (1.23) whose coefficients satisfy relations (1.26) are

referred to as orthogonal transformations. The orthogonality conditions (1.26) imply that

s sthe transpose of the matrix [ ] is equal to its inverse [ ] . That is,-1

1.1.5 Transformation of the Components of a Vector upon Rotation of theRectangular System of Axes to Which They Are Referred

Let us consider the vector a whose components relative to the rectangular systems of

i j i jaxes x(i = 1, 2, 3) and x ( j = 1, 2, 3) are a and a , respectively. That is,

Substituting relation (1.23) into (1.29), we obtain

Consequently,

Similarly, substituting relation (1.22) into (1.29), we get



Vectors 9

(1.33)

(1.34)

(1.35)

(1.36a)

Relations (1.31) and (1.32) can be written in matrix form as

or

and

or

1.1.6 Transformation of the Components of a Planar Vector upon Rotation of the

Rectangular System of Axes to Which They Are Referred

In two-dimensional (planar) problems we encounter certain vectors such as forces and

translations which act in the plane of the problem. In this section we establish the

relations between the components of such vectors referred to two sets of two mutually

perpendicular axes laying in the plane of the problem (see Fig. 1.6). Consider the vector

1 2F acting in the plane specified by the two mutually perpendicular axes x and x and

1 2denote its components with respect to these axes by F and F , respectively. Moreover,

1 2 1consider another set of two mutually perpendicular axes x and x located in the plane x

2 1 2 x and denote the components of vector F with respect to these axes by F and F ,

respectively. That is,

Referring to Fig. 1.6, we have

and



Cartesian Tensors10

(1.36b)

(1.37)

(1.38)

Figure 1.6 Components of a planar vector.

where

From relations (1.36a) and (1.36b) it can be seen that

1.1.7 Definition of a Vector on the Basis of the Law of Transformation of Its

Components

In Section 1.1, we have defined a vector as an entity possessing magnitude, direction

and sense and obeying certain rules. It may be represented in the three-dimensional space

by a directed line segment (arrow) whose direction is that of the vector. This definition

does not associate any system of axes with a vector and, thus, it is immediately apparent

that a vector has an existence independent of any system of axes in the same frame of

reference. On the basis of this definition, it was established in Section 1.1.5 that the

components of a vector transform, upon rotation of the rectangular system of axes to

which they are referred, in accordance with the transformation relation (1.31) or (1.32).

The reverse is also valid. That is, any entity defined with respect to any rectangular system

iof axes by an array of three numbers a (i = 1, 2, 3) satisfying relation (1.31) or (1.32) is

a vector



11Dyads

(1.39a)

(1.39b)

(1.39c)

(1.39d)

(1.40)

(1.41)

(1.45)

(1.44)

(1.43)

(1.42)

Thus, a vector may be defined as an entity which possesses the following properties:

1. With respect to any rectangular system of axes, it is specified by an array of three

i numbers a (i = 1, 2, 3) — its three Cartesian components.

2. Its Cartesian components referred to any two rectangular systems of axes are related

by the transformation relation (1.31) or (1.32).

1.2 Dyads

We define a dyad as the product a b of two vectors a and b which obeys the following

rules:

where is a real number. In general this product is not commutative. That is,

The following products are defined between a vector and a dyad, and between two

dyads:

On the basis of the above definition a dyad like a vector has an existence independent of

1 2any coordinate system. In what follows, we introduce a rectangular system of axes x , x ,

3 1 2 3 x specified by the orthogonal unit vectors i , i , i in order to permit the use of well-known

mathematical procedures. Thus, in terms of the components of the two vectors a and b,

1 2 3referred to the orthogonal unit vectors i , i , i , using the rule (1.39d), a dyad ab may be



Cartesian Tensors12

(1.46)

(1.47)

(1.49)

(1.50)

written as

or

The sum of two dyads a b and c d is a dyad whose components are equal to the sum

of the components of the two dyads

1.3 Definition and Rules of Operation of Tensors of the Second Rank

A tensor of th e second rank, also known as dya dic, is a linear combination of a finite

number of dyads. We denote tensors of the second rank by modified letter symbols as A,B, e, etc. For example,

A = a b + c d + e f (1.48)

Referring to relations (1.46) to (1.48), it is apparent that any tensor of the second rank can

1 2 3 be written with respect to the orthogonal unit vectors i , i , i as

or

ijThe nine quantities A (i, j = 1, 2, 3) are referred to as the cartesian components of the

1 2 3tensor of the second rank with respect to the set of the orthogonal unit vectors i , i , i or

1 2 3with respect to the rectangular system of axes x , x , x specified by these unit vectors.

Any tensor of the second rank can be specified by giving its nine cartesian components

1 2 3 11 22with respect to a set of orthogonal unit vectors i , i , i . The cartesian components A , A ,



Definition and Rules of Operations of Tensors of the Second Rank 13

(1.51)

(1.52c)

(1.52d)

33 A of a tensor of the second rank A are referred to as its diagonal components, while the

12 13 21 23 31 32cartesian components A , A , A , A , A , A of the tensor of the second rank A are

referred to as its non-diagonal components. On the basis of the foregoing presentation,

it is evident that the non-diagonal cartesian components of a tensor of the second rank Ai jare associated with two orthogonal directions (i and i where i j), whereas the diagonal

cartesian components of a tensor of the second rank are associated with a single direction

i j i itwice (i and i where i = j). Notice that the cartesian components of a vector a (a = i a)

are associated solely with one direction. A vector is a tensor of the first rank.

The sum of two tensors of the second rank is a tensor of the second rank whose

components referred to a rectangular system of axes are equal to the sum of the

components of the two tensors referred to the same system of axes. That is,

or

or

Tensors of the second rank obey the following rules:

A + B = B + A (1.52a)

( A + B) + D = A + ( B + D) (1.52b)

where and are real numbers. Moreover, the products of a tensor of the second rank and a vector obey the following rules:

A v v A (1.53a)

(A + B) v = A v + B v (1.53b)

A (a + b) = A a + A b (1.53c)

The dot product of a tensor of the second rank A and a vector a is a vector c whose

components can be established from those of the tensor A and the vector a as follows:



Cartesian Tensors14

(1.54)

(1.55)

(1.56)

(1.57)

(1.58a)

or

and

or

The dot products of a tensor of the second rank by a vector (1.54) and (1.55) may also

be obtained by multiplication of the matrices of their cartesian components referred to the

same rectangular system of axes. Thus,

and

where




(1.58b)

(1.59)

(1.61)

(1.62)

(1.64a)

(1.64b)

(1.65)

ijFrom relation (1.50), we may deduce that, in general, the cartesian components A of

a tensor of the second rank are given by

Moreover, the diagonal component of a tensor of the second rank A in the direction

n n1 1 n2 2 n3 3specified by the unit vector i = i + i + i is given as

nn n n A = i A i (1.60)

This relation may be rewritten in matrix form as

where [ A] is given by relation (1.58a) and {n} is equal to

The non-diagonal component of a tensor of the second rank A in the directions specified

n n1 1 n2 2 n3 3 s s1 1 s2 2 s3 3 by the orthogonal unit vectors i = i + i + i and i = i + i + i , is given

as

ns n s A = i A

i (1.63a)

sn s n A = i A i (1.63b)

ns snwhere the components A and A are not necessarily equal. Relations (1.63) may be

written in matrix form as

where



Cartesian Tensors16

(1.66)

(a)

As in the case of vectors, tensors of the second rank may be represented as follows:

1. By the symbolic representation using modified letter symbols, as A, B, e, . This

representation does not require a choice of a coordinate system.

2. By their cartesian components with respect to a set of unit orthogonal base

vectors, using

ij ij ij ij ij ij ij ij ij(a) Indicial notation as A , B , a , b , e , , , ,

(b) Matrix notation, that is,

n 1 2Example 1 Consider the two mutually perpendicular unit vectors i = 3/5i 4/5i and

s 1 2i = 4/5i + 3/5i . Moreover, consider the tensor

nn nsFind the components of the tensor A and A .

Solution Referring to relation (1.60), we have

nn n s

A = i A i (b)

Where

1 1 1 2 1 3 2 1 2 2 2 3 3 1 3 2A = 2i i + 3i i + i i + 4i i + 2i i + 3i i 2i i + i i

Thus,

and




(1.68)

The same results may be obtained using matrix multiplication. That is,

and

1.3.1 Example of a Tensor of the Second Rank

In this section we give an example of a quantity which is a tensor of the second rank.

Consider a body subjected to external forces. In general the particles of this body will

be stressed. In Section 2.8 we show that the state of stress of a particle of a body is

completely specified by a tensor of the second rank, which referring to Fig. 1.7, it is givenas

(1.67)

ij is called the stress tensor and (i, j = 1, 2, 3) are its components with respect to the

1 2 3system of orthogonal axes x , x , x . Notice than in order to specify a component of stress

we must give two directions in addition to its magnitude, namely, the direction of the

normal to the plane on which it acts and the direction of the component of stress. For

12example, in order to specify the component of stress , we must give its magnitude, the

1 2direction i of the normal to the plane on which it acts and direction i in which the

component of stress acts. The components of the tensor may also be presented in matrix

form as

In what follows we present an example.



Cartesian Tensors18

(b)

(a)

The components of stress acting

on the faces CC’EE’ and AA’DD’

are not shown.

Figure 1.7 Components of stress acting on a particle of a body.

Example 2 The state of stress at point A of the beam of rectangular cross section is

shown in Fig. b. Compute the components of stress acting on plane aa which is normal

1 3to the x x plane as shown in Fig. a.

Figure a Simply supported beam Figure b State of stress at pint A

of rectangular cross section. of the beam of Fig. a.

Solution Referring to Fig. b, the stress tensor at point A of the beam is

or

Referring to Fig. a, the unit vector normal to the plane aa is




(c)

(d)

nFigure c Components of stress acting on the plane normal to the unit vector i .

s n 1 3Moreover, the unit vector i normal to i and lying in the x x plane is

Thus,

and

n2 nThe component of stress acting on the plane normal to the unit vector i in the direction

2of the x axis is equal to

nThe components of stress acting on the plane normal to the unit vector i are shown in Fig.

c.



Cartesian Tensors20

(1.69)

(1.70)

(1.71a)

(1.71b)

(1.71c)

(1.71d)

(1.71e)

(1.71f)

1.4 Transformation of the Cartesian Components of a Tensor of the Second Rank

upon Rotation of the System of Axes to Which They Are Referred

i j ijConsider a tensor of the second rank /A and denote by A and A , its components with

i jrespect to the system of axes x(i = 1, 2, 3) and x ( j = 1, 2, 3), respectively. Thus, referring

to relation (1.49), we have

Referring to relation (1.59), we obtain

Substituting relation (1.22) into the above and using relation (1.59), we get

Thus,

This relation specifies the transformation of the components of a tensor of the second rank

upon rotation of the axes of reference. It can be expanded to give



Definition of a Tensor of the Second Rank on the Basis of the Law of Transforamtion 21

(1.71g)

(1.71h)

(1.71i)

(1.72)

(1.73a)

(1.73b)

(1.74)

Following a procedure analogous to the one employed in obtaining the transformation

relation (1.70), we may obtain

Relations (1.70) and (1.72) may be written in matrix form as follows

s swhere [ ] is the transformation matrix (1.24a) and [ ] is its transpose.T

1.5 Definition of a Tensor of the Second Rank on the Basis of the Law of

Transformation of Its Components

In Section 1.3 a tensor of the second rank was defined without referring to any system

of axes. Thus, it is explicitly apparent that a tensor has an existence independent of the

choice of the system of axes. The system of axes (specified by the orthogonal unit vectors

1 2 3i , i , i ) has been introduced subsequently in order to permit the use of well-known

mathematical procedures. On the basis of this definition of a tensor of the second rank,

it was established in Section 1.4 that its cartesian components transform upon rotation of

the right-handed system of axes to which they are referred in accordance with the

transformation relations (1.70) or (1.72). The reverse is also valid; that is, any physical

entity defined with respect to any rectangular system of axes by an array of nine numbers

ij A (i, j = 1, 2, 3) which satisfy relation (1.70) or (1.72) specifies a tensor of the second rank

The transformation relation (1.70) or (1.72) is often used as the basis for the definition

of a tensor of the second rank as an entity which possesses the following properties:

1. With respect to any set of rectangular axes it is specified by an array of nine

ijnumbers A (i, j = 1, 2, 3) — its nine cartesian components.



Cartesian Tensors22

(1.75)

(1.76)

(1.77)

2. Its cartesian components referred to any two right-handed rectangular system of

1 2 3 1 2 3axes specified by the orthogonal unit vectors i , i , i and i, i, i are related by the

transformation relation (1.70) or (1.72).

This definition of a tensor of the second rank has the shortcoming that it is dependent

upon the choice of a system of axes, while the definition presented in Section 1.3 is not.

1.6 Symmetric Tensors of the Second Rank

A symmetric tensor of the second rank is one whose components satisfy the following

relation:

For instance, the tensor /A whose components with respect to a rectangular system of axes

is given as

is a symmetric tensor of the second rank. If a tensor A is symmetric, we have

In Section 1.9 we show that for any symmetric tensor of the second rank, there exists at

1 2 3least one system of rectangular axes x , x , x , called principal , with respect to which the

diagonal components of the tensor assume their stationary values. That is, one of them

is a maximum and another is a minimum of the diagonal components of the tensor in any

direction. Moreover in Section 1.9, we show that, with respect to the principal axes, the

non-diagonal components of the tensor vanish. That is, the tensor assumes the following

diagonal form:

1 2 3In this case the diagonal components A , A and A are called the principal components

of the tensor .

1.7 Invariants of the Cartesian Components of a Symmetric Tensor of the SecondRank

Consider a tensor of the second rank A and denote its components with respect to the



Stationary Values of a Function Subject to a Constraining Relation

23

(1.79)

(1.80)

(1.81)

(1.82)

1 2 3 1 2 3 ik jmrectangular systems of axes x , x , x and x, x, x by A (i, k = 1, 2, 3), A ( j, m = 1, 2, 3).

Adding relations (1.71a), (1.71e) and (1.71i) and using relations (1.25), we obtain

1 11 22 33 11 22 33 II = A + A + A = A + A + A (1.78)

That is, the sum of the three diagonal components of a tensor of the second rank is

independent of the system of axes to which the components are referred. That is, it is

invariant to the rotation of the axes of reference. It is referred to as the first invariant of

the tensor . Moreover, by appropriate manipulation of relations (1.71) it may be shown

that a symmetric tensor of the second rank has the following invariants:

2 3where II and II are referred to as the second and third invariants of the tensor of the

second rank , respectively.

1.8 Stationary Values of a Function Subject to a Constraining Relation

1 2 3Consider a function f ( x , x , x ) having continuous first derivatives in a region R. We

1 2 3 1 2 3say that the function f ( x , x , x ) assumes a stationary value at a point P ( x , x , x ) in the

region R if the following relation is satisfied at this point:

A stationary value of a function could be a maximum or a minimum or a saddle point.

1 2 3If the variables x , x , x are independent, then a necessary and sufficient condition for the

satisfaction of relation (1.81) is

In many problems in mechanics, it is necessary to establish the stationary values of a

1 2 3 1 2 3function f ( x , x , x ), when the variables are related by a constraining relation g ( x , x , x )

1 2 3 1= 0. If the relation g ( x , x , x ) = 0 can be solved for one of the variables, say x , in terms1of the other two, then the resulting expression may be substituted into the function f ( x ,

2 3 x , x ) and a function is obtained. In this case it is apparent that the stationary

1 2 3 1 2 3values of f ( x , x , x ) under the constraining relation g ( x , x , x ) = 0 may be obtained from



Cartesian Tensors24

(1.83)

(1.84)

(1.85)

(1.86)

(1.87)

(1.88)

the following relation:

2 3Inasmuch as dx and dx are independent, the necessary and sufficient condition for the

satisfaction of the above relation is

1 2 3However, in certain problems, g ( x , x , x ) = 0 is a complicated function. In this case in

1 2 3order to establish the stationary values of f ( x , x , x ), it is convenient to use the ingenious

method proposed by Lagrange, which we describe in the sequel.

1 2 3 1 2 3Inasmuch as the variables x , x , x are related by the constraining relation g ( x , x , x )

i= 0, the increments dx (i = 1, 2, 3) in relation (1.81) are not independent. They are related

by the following relation:

3Solving equation (1.85) for dx , we obtain

3Substituting dx from the above relation into equation (1.81), we get

1 2Since dx , dx are independent, their coefficients in the above relation must vanish.

Hence,

1 2 3 1 2 3Thus, the function f ( x , x , x ) assumes stationary values at the points P ( x , x , x ) of the

region R whose coordinates satisfy the following relation:



Stationary Values of a Function Subject to a Constraining Relation

25

(1.89)

(1.90)

(1.91)

(1.92)

(1.93)

(a)

Consider the function

This function assumes its stationary values at points whose coordinates satisfy the

following relation:

1 2 3 1 2 3Inasmuch as the function F ( x , x , x , ) is not subjected to a constraint, dx , dx , dx and

d are independent and relation (1.91) is satisfied when

Substituting relation (1.90) into the above relations, we get

1 2These equations are identical to equations (1.89). Thus, the stationary values of f ( x , x ,

3 1 2 3 1 x ) subjected to the constraining relation g ( x , x , x ) = 0 and the stationary values of F ( x ,

2 3 x , x ) without a constraining relation occur at the same points. Consequently, the

1 2 3 1 2 3stationary values of f ( x , x , x ) subjected to the constraining relation g ( x , x , x ) = 0 and

the points at which they occur may be obtained from equations (1.92). The multiplying

constant is called the Lagrange multiplier .

Example 3 Using the method of Lagrange multipliers, find the point on the plane

specified by the following relation which is the nearest to the origin of the system of axes

1 2 3 x , x , x :

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Chapter 1 - Cartesian Tensors