Appendices

Appendix-1(A) Vector Analysis-Some Important Vector Relations

1.1 The Concept of a Vector

In meteorology, some physical properties such as mass, time, temperature, pressure, etc., are measured and reported by their magnitudes only. They are called scalars. However, there are some others for which we need to specify a direction in addition to magnitude. These are called vectors. In the latter category, we may mention the position of a point, displacement, velocity, acceleration, force, etc. [In the appendices as well as in text, vectors are denoted by letters in bold-face.]

We illustrate the concept of a vector by taking the simple case of the position of a particle P in space. Let it be located at a distance r from a fixed point O at time t (see Fig. 1.1', left panel). The position of P is then defined by a vector R = r r, where r is the magnitude of R, indicated by |R|, and r a unit vector in the direction from O to P, indicated by arrow. Here, both magnitude and direction are important to define the vector.

It is possible to resolve the position vector R into component vectors along the co-ordinate axes of a system of reference, by defining unit vectors along the coordinate axes.

Let i, j, k be the unit vectors along the axes of a rectangular co-ordinate system and x, y, z the resolved components of r along the axes OX, OY, OZ respectively. Then we can express the vector R as

where r =(x2 + y2 + z2)1/2, and r, x, y, z are all scalars.

A unit vector has a magnitude unity in a specified direction and plays an important role in vector analysis. It helps to provide the direction in which a vector acts. As we shall see in several parts of this book, the use of a vector is advantageous for two important reasons: firstly, it greatly simplifies mathematical treatment and secondly,

Fig. 1.1' The concept of a vector and its resolution into components

it provides a simpler physical and geometrical representation of mathematical results. Some of the standard mathematical operations with vectors are summarized in the following sections.

1.2 Addition and Subtraction of Vectors: Multiplication of a Vector by a Scalar

We have already demonstrated in (1.1') how the vectors are added. In general, if two vectors A and B are added, their sum C is a vector given by the vector equation

The sum C is obtained by laying off the vector B from the end-point of A and then drawing a vector from the initial point of A to the end-point of B (see Fig. 1.2').

A vector equation is equivalent to three scalar equations, since, as we saw in (1.1'), a vector is represented by its three components along the co-ordinate axes and two vectors are equal if their components are respectively equal. The geometric sum is commutative, like an ordinary arithmetic sum, i.e., it does not depend upon the order in which the vectors are added. Further, a vector sum is also associative, i.e., it does not matter how the individual vectors are grouped. For example, if S is the sum of three vectors A, B and C,

The sum of two vectors having the same direction and sense is a vector the magnitude of which is the sum of the magnitudes of the two vectors, the direction remaining the same. If a vector is added n times, the vector can simply be multiplied by the number n.

A negative vector, e.g., —B has the same magnitude as B but opposite direction. Accordingly, the vector difference A — B may be written as A +(—B).

1.3 Multiplication of Vectors

Two vectors can be multiplied, either scalarly or vectorially. A scalar product is indicated by a bold dot sign (.) placed between them, whereas a bold cross sign (x) indicates a vector product.

(a) The scalar product of two vectors

The scalar product, also called the dot product, of two vectors, A and B, is a scalar which is equal to the product of the magnitudes of the vectors and the cosine of the angle between them (Fig. 1.3'). Thus,

where A and B are respectively the magnitudes |A| and |B| of the vectors and a is the angle (< 180°) between them.

Fig. 1.3' The scalar product of two vectors

The relation (1.4') means that when two vectors are equal and parallel to each other, the angle between them is zero and cos a = 1. The scalar product is then simply the product of the magnitudes of the two vectors. But when they are at right angles to each other, cos 90° = 0, and the scalar product is zero. This is an important result, for, when applied to the unit vectors, i, j, k, we get i.i = j.j = k.k =1 (for a = 0); and, i.j = j.k = k.i = 0 (for a = 90) (1.5')

From (1.5') it follows that if we take the resolved components of two vectors A and B along the co-ordinate axes, their scalar product may be written in terms of the components as

(b) The vector product of two vectors

The vector product, also called the cross product, of two vectors, A and B, is a vector P the direction of which is that of a right-hand screw, i.e., perpendicular to the plane determined by the vectors A and B, and the magnitude of which is equal to the area of the parallelogram formed by them, i.e., equal to AB sin a, where a(< 180) is the angle between them (Fig. 1.4').

Fig. 1.4' The vector product of two vectors

The stipulation of a right-hand screw as determining the direction of the vector product denies it the property of commutativity possessed by a scalar product; for, if we reverse the order of the vectors, then the turning of B towards A, in the sense of a right-hand screw, would give the vector product a direction opposite to that of P. Thus,

The most important property of the vector product is that of its distributivity with respect to addition. This means that

Ax(B + C + D + •••)=AxB + AxC + AxD+ ... (1.8')

The proof of (1.8') is a little cumbersome and will not be attempted here.

If the vectors are parallel to each other, sin a = 0, and the vector product is zero. If they are perpendicular to each other, sin a = 1, and the magnitude of the product vector is simply the arithmetic product of the individual magnitudes of the vectors, the direction being at right angles to the plane of the two vectors according to the right-hand screw rule. We can apply the vector product definition given above immediately to the unit vectors and obtain ixi = jxj = kxk = 0. (since sin a = 0); and ixj = —jxi = k; jxk = —kxj = i; kxi = —ixk = j (19')

Using these relationships, we can obtain the vector product of two vectors A and B in terms of the products of their components as follows:

= (AyBz — AzBy)i +(AzBx — AxBz)j + (AxBy — AyBx )k (1.10' )

The result (1.10') can also be expressed in the determinant form i j k

Bx By Bz

(c) Multiple vector/scalar products

The methods of determining scalar/vector products of vectors described above may be applied successively in multiple products of vectors. The following are some of the products which occur frequently in several branches of science.

1.4 Differentiation of Vectors: Application to the Theory of Space Curves 337 If A, B, C are three vectors, then

A • BxC = B • CxA = C • AxB = -A • CxB = -B • AxC = -C • BxA (1.127)

The products in (1.127) may also be expressed in terms of the components of the vectors. For example,

A • BxC=(Axi + Ayj + Azk) • {(ByCz - BzCy)i +(BzCx - BxCz)j+ ,)

(1.137) may also be written in the form of a determinant

Ax

Ay

Az

Bx

By

Bz

Cx

Cy

Cz

It is easy to see that all the multiple products in (1.12/) represent the volume of a parallelepiped formed by the three vectors as contiguous sides.

The vector product of a vector with the vector product of two other vectors is represented as follows:

1.4 Differentiation of Vectors: Application to the Theory of Space Curves

Let the position vector r of a point P on a space curve be a function of the length of the arc s, measured from an initial point on the curve (Fig. 1.5').

Then, |Ar| is identical with As, and in the limit, as As ^ 0, Ar/As is a vector of length 1 directed along the tangent to the curve. Let us denote this unit vector by t. Then t = dr/ds (1.15')

The scalar products of vectors are differentiated just as ordinary scalar functions, regardless of the order of the vectors. In the case of vector products, however, the order of the vectors is important. Thus, if we differentiate the scalar and the vector products of two vectors, A and B, with respect to a scalar u, we get d(A *B)/du = (dA/du)B + A -(dB/du)

If a vector A is of constant length A, but changes direction only, then the differentiation of the scalar product A • A, with respect to a scalar u gives d(A -A)/du = dA2/du = 2A-(dA/du) = 0 (1.18')

Since neither the vector nor the derivative is to disappear, it follows from (1.18') that the derivative of a vector of constant length is at right angles to the vector itself. It is easy to see that if the length is constant, the end-point of the vector will move on a sphere. If the increment is infinitesimal, it is tangent to the sphere and thus perpendicular to the vector.

1.5 Space Derivative of a Scalar Quantity. The Concept of a Gradient Vector

Several scalar quantities, such as temperature, pressure, density, etc., vary in space and are continuous functions of space variables. We can express the space derivative of such a scalar by a vector which is called the gradient of that scalar quantity. To see how this is done, let us consider a pressure field around a pressure center, O. Since the field is continuous, we can locate surfaces of equal pressure, known as isobaric surfaces, at different distances from the center of the pressure. Let r be the position vector of a point P on one of these surfaces where the pressure is p (Fig. 1.6')

Then, in Cartesian co-ordinates, we can write dr = dx i + dy j + dz k

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