Chapter 1 Vector and Dyadic Algebra 1-1 Representations of Vector Functions A vector function has both magnitude and direction. The vector functions that we encounter in many physical problems are, in general, functions of space and time. In the first five chapters, we discuss only their characteristics as functions of spatial variables. Functions of space and time are covered in Chapter 6, dealing with a moving surface or a moving contour. A vector function is denoted by F. Geometrically, it is represented by a line with an arrow in a three-dimensional space. The length of the line corresponds to its magnitude, and the direction of the line represents the direction of the vector function. The convenience of using vectors to represent physical quantities is illustrated by a simple example shown in Fig. 1-1, which describes the motion of a mass particle in a frictionless air (vacuum) against a constant gravitational force. The particle is thrown into the space with an initial velocity Vo, making an angle 0o with respect to the horizon. During its flight, the velocity function of the particle changes both its magnitude and direction, as shown by Vi, v2, and so on, at subsequent locations. The gravitational force that acts on the particle is assumed to be constant, and it is represented by F in the figure. A constant vector function means that both the magnitude and the direction of the function are constant, being independent of the spatial variables, x and z in this case. The rule of the addition of two vectors a and b is shown geometrically by Fig. l-2a, b, or c. Algebraically, it is written in the same form as the addition of 2 Vector and Dyadic Algebra Chap. 1 t v- V2 /( F jV^3 / V \ 0 >x *o z = JCI 1 — — tan80 XQ = — vQ sin 2G0 Figure 1-1 Trajectory of a mass particle in 8 a gravitational field showing the velocity v and the constant force vector F at different g = gravitational constant locations. b b b (a) (b) (c) Figure 1-2 Addition of vectors, a -f- b = c. two numbers of two scalar functions, that is, c = a + b. (1.1) The subtraction of vector b from vector a is written in the form d = a-b. (1.2) Now, — b is a vector that has the same magnitude as b, but of opposite direc- tion; then (1.2) can be considered as the addition of a and (—b). Geometrically, the meaning of (1.2) is shown in Fig. 1-3. The sum and the difference of two vectors obey the associative rule, that is, a+b=b+a (1.3) and a-b = -b + a. (1.4) They can be generalized to any number of vectors. The rule of the addition of vectors suggests that any vector can be considered as being made of basic components associated with a proper coordinate system. The Sec. 1-1 Representations of Vector Functions 3 -b b Figure 1-3 Subtraction of vectors, a - b = d. most convenient system to use is the Cartesian system or the rectangular coordinate system, or more specifically, a right-handed rectangular system in which, when JC is turning to y9 a right-handed screw advances to the z direction. The spatial variables in this system are commonly denoted by x, y, z. A vector that has a magnitude equal to unity and pointed in the positive x direction is called a unit vector in the x direction and is denoted by x. Similarly, we have y, z. In such a system, a vector function F that, in general, is a function of position, can be written in the form F=Fxx + Fyy+F2z. (1.5) The three scalar functions Fx, Fy, Fz are called the components of F in the direction of Jc, y, and z, respectively, while Fxx, Fyy, and Fzz are called the vector components of F. The geometrical representation of F is shown in Fig. 1-4. It is seen that Fx, Fy, and Fz can be either positive or negative. In Fig. 1-4, Fx and Fz are positive, but Fy is negative. z i i S ^\ Fzz F ^\ /^ p *+—" "Fxx J **y X Figure 1-4 Components of a vector in a Cartesian system. In addition to the representation by (1.5), it is sometimes desirable to express F in terms of its magnitude, denoted by |F|, and its directional cosines, that is, F = |F| (cos ax + cos $y + cos yz). (1.6) 4 Vector and Dyadic Algebra Chap. 1 a, P, and y are the angles F makes, respectively, with jc, y, and z, as shown in Fig. 1-4. It is obvious from the geometry of that figure that IFI^^ + F^ + F^)172 (1.7) and cosa = |[' cosP = M- cost = W\- (L8) Furthermore, we have the relation cos2 a + cos2 P + cos2 y = 1. (1.9) In view of (1.9), only two of the directional cosine angles are independent. From the previous discussion, we observe that, in general, we need three parameters to specify a vector function. The three parameters could be Fx, Fy, and Fz or |F| and two of the directional cosine angles. Representations such as (1.5) and (1.6) can be extended to other orthogonal coordinate systems, which will be discussed in a later chapter. 1-2 Products and Identities The scalar product of two vectors a and b is denoted by a • b, and it is defined by a b = |a| |b|cosGf (1.10) where 6 is the angle between a and b, as shown in Fig. 1-5. Because of the notation used for such a product, sometimes it is called the dot product. By applying (1.10) to three orthogonal unit vectors u\, W2, W3, one finds */•*; = { I] ]^Jj )> 1,7 = 1,2,3. (l.ll) The value of a • b can also be expressed in terms of the components of a and b in any orthogonal system. Let the system under consideration be the rectangular system, and let c = a — b; then |c|2 = |a - b|2 = |a|2 + |b|2 - 2 |a| |b| cos 0. Hence |a|2 + |b|2 |a b|2 a.b=|aHb|cose= 2- - 2 2 2 = al + aj + a\ + b x + b y + b z - (ax - bxf - (ay - b,f - (a, - bzf ~ 2 = axbx + ayby + a2bz. (1.12) y/^ j " ^ Figure 1-5 Scalar product of two vectors, • a a b = |a||b| cos 9. Sec. 1-2 Products and Identities 5 By equating (1.10) and (1.12), one finds cos6 = j^M*+«A + «A) (1.13) = cos aa cos at, + cos (3fl cos P*> -f cos ya cos jb, a relationship well known in analytical geometry. Equation (1.12) can be used to prove the validity of the distributive law for the scalar products, namely, (a + b) • c = a • c + b • c. (1.14) According to (1.12), we have (a + b) • c = (ax + bx) cx + (ay + by) cy + (az + bz) cz = {axcx + aycy + azcz) + (bxcx + bycy + bzcz) = a • c + b • c. Once we have proved the distributive law for the scalar product, (1.12) can be verified by taking the sum of the scalar products of the individual terms of a and b. The vector product of two vector functions a and b, denoted by a x b, is defined by axb= |a||b|sin6Mc, (1.15) where 0 denotes the angle between a and b, measured from a to b; iic denotes a unit vector perpendicular to both a and b and is pointed to the advancing direction of a right-hand screw when we turn from a to b. Figure 1-6 shows the relative position of uc with respect to a and b. Because of the notation used for the vector product, it is sometimes called the cross product, in contrast to the dot product or the scalar product. For three orthogonal unit vectors in a right-hand system, we have wi x«2 = W3, M2 x "3 = i/i, and W3 x u\ — it2> It is obvious that«/ x fi,- = 0, i = 1, 2, 3. From the definition of the vector product in (1.15), one finds bxa = -axb. (1.16) The value of a x b as described by (1.15) can also be expressed in terms of the components of a and b in a rectangular coordinate system. If we let a x b = v = axb i i uc n ^* Figure 1-6 Vector product of two vectors, yr j ° a a x b = |a||b| sin0«c; ttc ± a, uc JL b. — • 6 Vector and Dyadic Algebra Chap. 1 vxx + vyy + vzz, which is perpendicular to both a and b, then a • v = axvx + ciyVy + azvz = 0, (1.17) b v = bxvx + byVy + bzvz = 0. (1.18) Solving for vx/vz and vy/vz, from (1.17) and (1.18) we obtain Vx z a _ **>^ ~~ zby Vy_ __ azfcjc — Clxbz vz axby - a^ft*' vz axby - ^6* Thus, VX Vy VZ a aybz — azby cizbx ~~ tf*^z &xby ~~ ybx Let the common ratio of these quantities be denoted by c, which can be determined by considering the case with a = Jc, b = j>; then v = axb = z; hence from the last ratio, we find c = 1 because vz = 1 and ax — by = 1, while ay = bx = 0. The three components of v, therefore, are given by vx = tf^ — azby vy = azbx-axbz , (1.19) vz = axby — aybx which can be assembled in a determinant form as x y z v= ax ay az .