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A REVIEW OF VECTOR AND
MATRIX ALGEBRA USING
SUBSCRIPTISUMMATION
CONVENTIONS
This chapter presents a quick review of vector and matrix algebra. The intent is not to cover these topics completely, but rather use them to introduce subscript notation and the Einstein summation convention. These tools simplify the often complicated manipulations of linear algebra.
1.1 NOTATION
Standard, consistent notation is a very important habit to form in mathematics. Good notation not only facilitatescalculationsbut, like dimensionalanalysis, helps to catch and correct errors. Thus, we begin by summarizing the notational conventions that
will be used throughout this book, as listed in Table 1 .l.
TABLE 1.1. Notational Conventions Symbol Quantity a A real number A complex number A vector component A matrix or tensor element An entire matrix A vector @, A basis vector T A tensor
L An operator
1
2 A R E W W OF VECTOR AND MATRIX ALGEBRA
A three-dimensionalvector can be expressed as
v = VX& + VY&, + VZ&, (1.1)
where the components (Vx, V,, V,) are called the Cartesian components of and (ex.e,, $) are the basis vectors of the coordinate system. This notation can be made more efficient by using subscript notation, which replaces the letters ( x , y, z ) with the numbers ( 1 , 2 , 3 ). That is, we define:
Equation 1.1 becomes
or more succinctly,
i= 1,2,
Figure 1.1 shows this notational modification on a typical Cartesian coordinate sys- tem. Although subscript notation can be used in many different types of coordinate systems, in this chapter we limit our discussion to Cartesian systems. Cartesian basis vectors are orthonormal and position independent. Orthonoml means the
magnitude of each basis vector is unity, and they are all perpendicular to one another.
Position independent means the basis vectors do not change their orientations as we move around in space. Non-Cartesian coordinate systems are covered in detail in Chapter 3. Equation 1. 4 can be compactedeven further by introducingthe Einstein summation convention, which assumes a summation any time a subscript is repeated in the same term. Therefore,
i=1,2,
I I I Y I
Figure 1.1 The Standard Cartesian System
