Analytical Tools

A. l Linear Algebra

In this book no formal distinction is made between vectors and matrices, the former being simply column matrices, as is common in treatments of linear algebra. In par­ticular the familiar vectors of mechanics, such as force and velocity, are simply three – component column matrices. We use boldface letters for both matrices and vectors, for example, A = [a, y] and v = [u,]. The corresponding lowercase letter defines the magnitude (or norm) of the vector. The transpose and inverse are denoted as usual by superscripts, for example, AT and A-1. When appropriate to the context, a subscript is used to denote the frame of reference for a physical vector, for example, V£ = [u v wr denotes a vector whose components in frame FE are (и, v, w). The three – component vectors of physics have the following properties:

Scalar product

c = a • b = arb = + a2b2 + a3b3 (A.1,1)

c is a scalar, with magnitude ab cos в, where в is the angle between a and b

Vector product

(A. 1,2)

c is a vector perpendicular to the plane of a and b, with direction following the right-hand rule for the sequence a, b, c and has the magnitude ab sin в, where в is the angle (< 180°) between a and b

Unit vectors

The basis unit vectors are i, j,k such that

(A. 1,4)


^ ‘


where x(O) is the value of x(t) when t = 0.1 The process may be repeated to find the higher derivatives by replacing x(t) in (A.2,2) by x(t), and so on. The result is

dn~^x dn~2x

= ~ (0) – s (0)—————— VO) + snx(s) (A.2,3)


Let the integral be

and let it be required to find y(s). By differentiating with respect to t, we get

= x(t)



1 1

y(s) = — x(s) + — y(0)