x2

v1

x1

FIGURE 2.1 Vector components.

and notation that will be used throughout the text. Most readers will have

a degree of familiarity with the concepts described herein and may need to

come back to this section only if clari¬cation on the notation used farther

on is required. The topics will be presented in terms of Cartesian coordinate

systems.

2.2.1 VECTORS

Boldface italic characters are used to express points and vectors in a three-

dimensional Cartesian space. For instance, e1 , e2 , and e3 denote the three

unit base vectors shown in Figure 2.1, and any given vector v can be ex-

pressed as a linear combination of these vectors as,

3

v= vi ei (2.1)

i=1

In more mathematical texts, expressions of this kind are often written with-

out the summation sign, , as,

v = vi ei (2.2)

Such expressions make use of the Einstein or summation convention, whereby

the repetition of an index (such as the i in the above equation) automatically

implies its summation. For clarity, however, this convention will rarely be

used in this text.

3

2.2 VECTOR AND TENSOR ALGEBRA

The familiar scalar or dot product of two vectors is de¬ned in such a

way that the products of the Cartesian base vectors are given as,

ei · ej = δij (2.3)

where δij is the Kroneker delta de¬ned as,

1 i=j

δij = (2.4)

0 i=j

Because the dot product is distributive with respect to the addition, the

well-known result that the scalar product of any two vectors u and v is

given by the sum of the products of the components is shown by,

3 3

u·v = ui ei vj ej

·

±=1 j=1

3

= ui vj (ei · ej )

±,β=1

3

= ui vi = v · u (2.5)

±=1

An additional useful expression for the components of any given vector v

follows from Equation (2.3) and the scalar product of Equation (2.1) by ej

to give,

vj = v · ej ; j = 1, 2, 3 (2.6)

In engineering textbooks, vectors are often represented by single-column

matrices containing their Cartesian components as,

®

v1

[v] = ° v2 » (2.7)

v3

where the square brackets symbols [ ] have been used in order to distinguish

the vector v itself from the single-column matrix [v] containing its Cartesian

components. This distinction is somewhat super¬‚uous unless more than one

basis is being considered. For instance, in the new basis e1 , e2 , and e3

shown in Figure 2.2, the same vector v will manifest itself with di¬erent

components, in which case the following notation can be used,

®

v1

[v] = ° v2 » (2.8)

v3

4 MATHEMATICAL PRELIMINARIES

x3

Q

0

x3

0

x2

e3

0

e2

0

e3

Q

e2

e1

x2

0

e1

x1

Q

0

x1

FIGURE 2.2 Transformation of the Cartesian axes.

It must be emphasized, however, that although the components of v are

di¬erent in the two bases, the vector v itself remains unchanged, that is,

3 3

v= vi ei = vi ei (2.9)

±=1 ±=1

Furthermore, the above equation is the key to deriving a relationship be-

tween the two sets of components. For this purpose, let Qij denote the dot

products between the two bases as,

Qij = ei · ej (2.10)

In fact, the de¬nition of the dot product is such that Qij is the cosine of the

angle between ei and ej . Recalling Equation (2.6) for the components of a

vector enables the new base vectors to be expressed in terms of the old, or

vice-versa, as,

3 3

ej = (ej · ei ) ei = Qij ei (2.11a)

±=1 ±=1

3 3

ei = (ei · ej ) ej = Qij ej (2.11b)

j=1 j=1

Substituting for ei in Equation (2.6) from Equation (2.11b) gives, after

5

2.2 VECTOR AND TENSOR ALGEBRA

simple algebra, the old components of v in terms of the new components as,

vi = v · ei

3

=v· Qij ej

j=1

3 3

= Qij (v · ej ) = Qij vj (2.12a)

j=1 j=1

A similar derivation gives,

3

vi = Qji vj

j=1

The above equations can be more easily expressed in matrix form with the

help of the 3 — 3 matrix [Q] containing the angle cosines Qij as,

[v] = [Q][v] (2.13a)

[v] = [Q]T [v] (2.13b)

where,

®

e1 · e1 e1 · e2 e1 · e3

[Q] = ° e2 · e1 (2.14)

e2 · e2 e2 · e3 »

e3 · e1 e3 · e2 e3 · e3

As a precursor to the discussion of second-order tensors, it is worth

emphasising the coordinate independent nature of vectors. For example, the

vector equations w = u+v or s = u · v make sense without speci¬c reference

to the basis used to express the components of the vectors. Obviously, a

vector will have di¬erent components when expressed in a di¬erent basis”

but the vector remains unchanged.

6 MATHEMATICAL PRELIMINARIES

EXAMPLE 2.1: Vector product

0

x3 x3

v

Q

uv

+

0

x2

45“

x2

u

45“

x1

0

x1

®

1 ’1 0

1

[Q] = √ ° 1 1 √0 »

20 0 2

®

1

[u] = ° 2 » ;

0

®

3

1°»

T

[u] = [Q] [u] = √ 1

20

® ®

1

0

1

[v] = [Q]T [v] = √ ° √ »

1

[v] = ° 1 » ;

2

1 2

As an example of the invariance of a vector under transformation con-

sider the two vectors u and v and the transformation [Q] shown above.

The vector or cross product of u and v is a third vector u—v whose

components in any base are given as,

®

u2 v3 ’ u3 v2

[u—v] = ° u3 v1 ’ u1 v3 »

u1 v2 ’ u2 v1

We can apply this equation in both systems of coordinates and obtain

a di¬erent set of components as,

® ®

1

2

1

[u—v] = √ ° √ »

[u—v] = ° ’1 » ; ’3

2

1 2

7

2.2 VECTOR AND TENSOR ALGEBRA

EXAMPLE 2.1 (cont.)

The fact that these two sets of components represent the same vector

u—v can be veri¬ed by checking whether in accordance with (2.13a)

[u—v] = [Q][u—v] . This is clearly the case as,

® ® ®

1 ’1 0 1

2

1 1