# 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)

^ ‘ t=0 |

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)

TRANSFORM OF AN INTEGRAL

Let the integral be

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

= x(t)

thus

and

1 1

y(s) = — x(s) + — y(0)