Vector Analysis

INDEX

PART I

THEORY

CHAPTER 1

SCALARS AND VECTORS

1-1. Scalars

A quantity which can be represented by a single number is known as a scalar. The elementary charge on an electron is a particularly simple example of such a quantity. The speed of light in a vacuum is another.

Let f(x,y,z,t) be a single-valued function of the parameters x, y, z, and t within some domain of definition for f. To every admissible set of values for the parameters there corresponds a single number. Thus f(x,y,z,t) is a scalar function of x, y, z, and t. If x, y, and z are cartesian coordinates and t is time, the quantity f is a scalar function of position in space and of time. A familiar example of such a scalar function is the temperature within a material substance. In the most general case the temperature varies from point to point, and at each point it may vary with time. Likewise, both pressure and mass density within a region through which a compressible fluid is flowing depend upon position and time.

The number of parameters upon which a scalar function depends varies from case to case. Within an incompressible fluid at rest and under the influence of gravity, for example, the pressure depends solely upon depth below the surface. When a hard rubber object is charged by rubbing with fur, the density of charge in general varies from point to point on the surface of the object and is a function of the two parameters necessary to specify position on the surface. The temperature of the earth’s atmosphere, on the other hand, depends upon a rather large number of variables. Among these are the three coordinates of position, the time of day, the season of the year, and an unknown number of parameters needed to specify conditions in the sun.

Scalar functions are those with which one meets in the differential and integral calculus. It is assumed that the reader is familiar with such functions, and with the concepts of limit and continuity and the processes of differentiation and integration.

Finally, a region throughout which a scalar function is defined will often be referred to as a scalar field.

EXERCISES

1. Give examples of well-known mathematical scalar constants.

2. Give additional examples of physical scalar constants.

3. Give additional examples of scalar fields.

4. Consider the three-dimensional scalar field

φ = x² + 2y² + 3z²

where x, y, z are rectangular cartesian coordinates. What are the equipotential surfaces of φ; that is, what are the surfaces φ = constant?

5. Under the assumption that density ρ is proportional to pressure p, show that the pressure in a fluid at rest in the earth’s gravitational field is given by

where h is height above some reference level, p0 is pressure at the reference level, g is the acceleration of gravity, and K is an appropriate constant.

Step 1: The difference in pressure dp between height h and height h + dh is the weight of a column of air of unit cross-sectional area contained between the levels h and h + dh. Write a relation expressing this fact, noting that pressure decreases as height increases.

Step 2: Replace p by an expression in p.

Step 3: Integrate with respect to h.

6. In an ideal gas, pressure p, absolute temperature T, and density ρ are related by the equation of state

where R is the universal gas constant, and M is the molecular mass of the gas in question. It has been shown by rocket and other measurements that up to 70 miles altitude the pressure in the earth’s atmosphere is divided by 10 for every 10-mile increase in height. Assuming a constant atmospheric temperature (which is not a correct assumption), use the result of Exercise 5 to calculate an average temperature for the earth’s atmosphere from the ground to 70 miles altitude. Take g = 980.1 cm/sec², R = 8.314 × 10⁷ ergs/°K mole, and M = 28.97.

7. In Exercise 5 it was tacitly assumed that the acceleration of gravity g was constant. Actually g varies inversely as the square of the distance from the center of the earth. Let g0 be the value of g at the surface of the earth, and let h denote height above the ground. Taking the radius of the earth as 3,963 miles, write an expression for g in terms of g0 and h. Using this expression for g, rework Exercise 5 to obtain p as a function of height.

8. Using the result of Exercise 7, rework Exercise 6. How much error was introduced into the calculation of Exercise 6 by using the value of g existing at the earth’s surface?

1-2. Vectors

The essential feature of a scalar is its magnitude. Some quantities, however, possess a characteristic direction as well as magnitude. Among these, for example, are forces, velocities, accelerations, and current densities.

To specify such quantities it is necessary to state both the magnitude and the direction. Geometrically this can be done by representing the quantity in question as an arrow of suitable length and appropriate direction. Arithmetically such a quantity can be specified by three ordered numbers, for if the representative arrow be placed with its tail at the origin of a fixed coordinate system, then the three numbers comprising the coordinates of the arrow tip uniquely specify both the length and direction of the arrow. These three numbers may, of course, be the cartesian coordinates, spherical coordinates, cylindrical coordinates, or any other set of coordinates of the arrow tip.

It is a familiar fact that when two forces act upon a mass with a common point of application, their combined effect is that which would be produced by a single force of a uniquely determined magnitude and direction acting alone. This single resultant force can be obtained by applying the familiar parallelogram rule. Thus, if in Fig. 1-1 OA and OB represent two forces acting at a point 0, then the diagonal OR of the parallelogram, of which OA and OB are adjacent sides, represents the force which is the resultant of OA and OB acting together. The magnitude of the resultant force is represented by OR on the same scale as that used in drawing OA and OB, and the direction of the resultant force is that of OR. Both the magnitude and direction of OR can be calculated from the corresponding quantities for OA and OB graphically using the parallelogram construction, or analytically by simple trigonometric means.

FiG. 1-1. The parallelogram rule for addition of forces.

Quantities which possess magnitude and direction and which combine according to the parallelogram law are known as vectors.

It is important to note that all three of the properties listed above are essential parts of the vector concept. Analytical processes involving vector quantities must take into account the parallelogram law as well as the directional properties of vectors. Magnitude and direction alone do not make a vector quantity. For example, finite rotations of a body possess a magnitude, namely, the total angle of rotation, and a characteristic direction, namely, that of the axis of rotation. But a little reflection will show that the parallelogram law cannot be used in general to obtain the resultant of two such rotations performed in succession.

There is another fundamental point here which must be emphasized. Throughout this text vectors are thought of as physical or geometric entities. As such, each vector quantity has a meaning completely independent of any coordinate system which may be introduced for the purposes of calculation or mathematical analysis. One must use the physical or geometric meaning of any given vector quantity as the ultimate basis from which to derive a mathematical expression for the vector in terms of whatever coordinates are used. Also, in changing from one set of coordinates to another set, the physical or geometric meaning is unaltered by the transformation, even though the representative expression may change markedly. This invariance in physical or geometric meaning forms the basis from which to derive the mathematical transformation from the old to the new expression.

Often it is convenient, sometimes essential, to think of a vector as localized in space. A force vector, for example, is thought of as attached to the point of application of the physical force itself. Again, within a moving fluid the velocity of flow can be represented by what is called a field of vectors. With each point in the region of flow may be associated a vector representing the speed and direction of flow. In the general case this velocity vector varies from point to point in space and at each point may vary with time. Such vector fields are constantly met with in physics and engineering.

EXERCISES

1. Give additional examples of vectors and vector fields.

2. Prove, by presenting a suitable counterexample, that the finite rotations in space are not vector quantities.

3. Prove that the positions of points in space relative to a fixed origin are vector quantities. (Hint: Think of the positions as displacements from the origin, and show that such displacements combine according to the parallelogram law.)

1-3. Notation

It is customary to represent a vector quantity by means of a letter in boldface type. The boldface character signifies both the magnitude and the directional properties of the vector. Ordinary absolute value signs are often used to indicate the magnitude alone. Thus the vector A A . A simpler method, which will be used throughout this text, is to denote the magnitude by the italic form of the letter used in boldface to denote the vector quantity.

Often it will be convenient to speak of "points r, A, B, . . . ," meaning thereby the tips of the vectors r, A, B, . . . when the vectors are placed with their tails at the origin of coordinates. This convention will be adopted henceforth.

1-4. Addition of Vectors

Let A and B denote two vectors. The resultant vector obtained from A and B by the parallelogram rule is called the sum of A and B. The addition of two vectors is denoted in the usual fashion by A + B. But it is to be remembered that the plus sign in this expression calls for the operation of combining the two vectors by the parallelogram method and has, therefore, a different meaning from that of ordinary scalar addition.

The sum of two vectors A and B can be obtained geometrically by placing the tail of A at the tip of B as in Fig. 1-2. Then the arrow, the tail of which coincides with that of B and the tip of which coincides with that of A, represents the sum A + B. Similarly, the tail of B can be placed at the tip of A and the triangle completed to obtain A + B. Frequently it is convenient to visualize the addition of two vectors in terms of this triangle law rather than in terms of the parallelogram rule. Plainly the two methods are equivalent.

FIG. 1-2. The triangle law for addition of vectors.

Often in the course of an analysis it is desirable to resolve a vector into a sum of two or more other vectors. This can be done by a single or repeated application of the triangle law. Figure 1-2, which earlier was regarded as illustrating the addition of A and B to give A + B, can also be thought of as showing the resolution of a vector C into the two vectors A and B. Resolution of a vector V into the sum of four noncoplanar vectors is shown in Fig. 1-3.

FIG. 1-3. Resolution of a vector into four noncoplanar vectors.

The vector obtained by projecting a vector A orthogonally onto a line of specified direction is known as the component of A in the specified direction. In Fig.1-4 the component of A in the direction of line l is the vector Al. If θ is the acute angle between A and l, then the magnitude of Al is A cos θ. The resolution of a vector into mutually perpendicular components is of especial importance both in the plane and in three dimensions. In the former case the vector is resolved into two perpendicular components; and in the latter case into three mutually orthogonal components.

FIG. 1-4. The component of a vector along a direction different from that of the vector.

EXERCISES

In the following exercises let (x,y) and (x,y,z) denote rectangular coordinates in the plane and in space respectively.

1. A vector in the plane is three units long and makes an angle of 135° with the positive x axis. What are its components along the positive x and positive y directions?

2. A vector in space is four units long and makes angles of 45° and 60° with the x and y axes respectively. What is its component along the positive z direction?

3. A vector in the xy plane extends from (0,0) to the point (3,−4). Find its length and the angle it makes with the positive x axis.

4. A vector in space has the components (−3,4,−12). What is its length, and what angle does it make with the positive y axis?

5. A vector in the xy plane is 13 units long, and has the component −5 along the x axis. What is its direction?

6. A vector in space of unit length has the component ½ along the x axis and makes an angle of 120° with the positive z axis. What is its y component?

Do Exercises 7 to 15 graphically.

7. Vectors A and B in the xy plane have the components (5,4) and (−3,2) respectively. Obtain A + B.

8. A and B are unit vectors in the xy plane making angles 30° and 120° respectively with the positive x axis. Find A + B.

9. Vectors A and B in the xy plane have the components (3,−6) and (2,1). Resolve A into B plus a third vector C, obtaining the components of C.

10. A lies in the xy plane and has the components (7,−2). Find the component of A along the line y = x.

11. The true air speed of a plane is 150 knots; its heading is 35° east of north. The wind is 30 knots from the southeast. What are the plane’s speed and direction of motion over the ground?

12. A ship is steaming due south at 20 knots. The ocean currents, however, are moving to the northwest at 2 knots. In what direction, and at what speed in that direction, is the ship actually moving?

13. The true air speed of an airplane is 250 knots. At the level at which the plane is flying the wind is from north-northeast at 70 knots. What heading must the plane assume in order that its track over the ground be due east, and what will the plane’s speed over the ground be?

14. As the sun is setting, a train is moving along a straight track heading 20° west of south (S 20° W). To the east of the train is a straight wall running from S 10° E to N 10° W. If the train is going 70 miles/hr, how fast is the train’s shadow moving along the wall?

15. A mass of 20 g is sliding down a perfectly smooth inclined plane which makes an angle of 25° with the horizontal. (a) How fast is the mass accelerating? (b) How long after starting from rest will it be traveling at 10 m/sec? (Use Newton’s law: F = ma; F = force, a = acceleration, m = mass. Determine F from the fact that it is the component along the plane of the