C Useful Theoretical Approaches and
Formulas
C.l Vector Notations Used in the Book
In fluid mechanics, two quantities are distinguished; scalars and vectors. A
scalar is a real number or a quantity that has magnitude. For example, length,
temperature, and blood pressure are represented by numbers. On the other
hand, a vector usually is described as a quantity that has both, magnitude and
direction. For example, wind velocity is a vector quantity, and has both speed
and an associated direction. In this book, a vector is represented by an arrow,
il, or is written as a boldface symbol, u. The bold type u denotes velocity
at a specified time and position in space. To describe a vector analytically,
its projections (components) on the x, y and z axes of the selected coordinate
system are used. For example, in Fig. 2.1a, the components of vector u are u,
v and w. In general, for vector representation, the following notation is used:
u = (u, V, w).
(C.1)
In fact, the projections of vector u on the coordinate axes should also be
treated as vectors. Therefore, instead of notation (C.1), another representation
is frequently used, i. e.:
u = U2 + V] + wk,
(C.2)
or:
u = ui + vj + wk,
(C.3)
where i = (1,0,0), j = (0,1,0), and k = (0,0,1).
The vectors i, j and k are unit vectors that form a basis for the system
of three-dimensional vectors. For example, the two-dimensional vector a in
Fig. C.1 can be represented as a = 9i + 4k and vector b takes the form