CONTINUUM MECHANICS for ENGINEERS

FIGURE 2.1A
Unit vectors in the coordinate directions x 1, x 2, and x 3.
FIGURE 2.1B
Rectangular components of the vector v.
order dimension spaces play integral roles in continuum topics. Because a
scalar has only a single component, it will have the same value in every
system of axes, but the components of vectors and tensors will have different
component values, in general, for each set of axes.
In order to represent vectors and tensors in component form, we introduce
in our physical space a right-handed system of rectangular Cartesian axes
Ox1x2x3, and identify with these axes the triad of unit base vectors , , ê1 ê2 ê 3
shown in Figure 2.1A. All unit vectors in this text will be written with a
caret placed above the boldfaced symbol. Due to the mutual perpendicularity
of these base vectors, they form an orthogonal basis; furthermore, because
they are unit vectors, the basis is said to be orthonormal. In terms of this
basis, an arbitrary vector v is given in component form by
3
∑
v = veˆ + v eˆ + v eˆ 1 1 2 2 3 3 = vieˆi (2.2-1)
This vector and its coordinate components are pictured in Figure 2.1B. For
the symbolic description, vectors will usually be given by lowercase Latin
letters in boldfaced print, with the vector magnitude denoted by the same
letter. Thus v is the magnitude of v.
At this juncture of our discussion it is helpful to introduce a notational
device called the summation convention that will greatly simplify the writing
i=
1