Problems and Worked Solutions in Vector Analysis

By L.R. Shorter

"A handy book like this," noted The Mathematical Gazette, "will fill a great want." Devoted to fully worked out examples, this unique text constitutes a self-contained introductory course in vector analysis for undergraduate and graduate students of applied mathematics.
Opening chapters define vector addition and subtraction, show how to resolve and determine the direction of two or more vectors, and explain systems of coordinates, vector equations of a plane and straight line, relative velocity and acceleration, and infinitely small vectors. The following chapters deal with scalar and vector multiplication, axial and polar vectors, areas, differentiation of vector functions, gradient, curl, divergence, and analytical properties of the position vector. Applications of vector analysis to dynamics and physics are the focus of the final chapter, including such topics as moving rigid bodies, energy of a moving rigid system, central forces, equipotential surfaces, Gauss's theorem, and vector flow.

• Language: English
• Publisher: Dover Publications
• Release date: Jun 1, 2014
• ISBN: 9780486795485

Book preview

Index

CHAPTER I

ADDITION

[1].   Definition of a Vector.

A vector is a magnitude which can be represented by a straight line of finite length drawn in a definite direction.

In the following pages the term vector will be applied both to the magnitude itself and to its representation by a finite straight line drawn in a parallel direction.

Nothing is said in this definition of the point from which the vector starts—the initial point—so that we may consider the position of this point to be arbitrary.

There are, then, an infinite number of vectors having the common properties of a given finite length and a given direction. Hence all vectors having the same given length and the same direction are to be considered equivalent: thus a vector from any initial point may be replaced by an equivalent vector from any other initial point. The point which terminates a vector is called its terminal point or extremity.

Vectors in the text will be represented by clarendon letters, such as a, b, c, r; but when referred to in diagrams, they will be represented by large capitals in clarendon, thus AB, CD.

Finite algebraical quantities in which direction is not involved are called scalar quantities or scalars: these will be represented in the text in ordinary type.

Such quantities as Force, Acceleration, Velocity (when the direction of motion of the latter is taken into account), Fluid Flow in a definite direction, are examples of vectors.

Among scalars are included: the temperature of a body at a given point in it; the work done by forces on a body when it is displaced by their action from one position to another; the mass of a body; an interval of time.

[2].   Addition of Vectors.

The term addition is applied to a method of combining vectors, which is analogous to the ordinary addition of algebraical quantities, and which is familiar in the graphic composition and resolution of forces.

FIG. 1.

Let AB represent a vector, with initial point A and terminal point B (Fig. 1), and let another BC be drawn from the terminal point B of AB. The plus sign in clarendon type, +, will be employed to denote this operation. Calling the vector AB, a and the vector BC, b, the process just described is symbolised by a + b.

Now if we consider the point A as joined to C by the vector AC or c, we see that the process symbolised by a + b leads to the same point as that reached by drawing the vector c from A, so that we may say,

where = may be read as equivalent to. (See also next Section.)

This, then, is an example of the addition of vectors, and we conclude from it, that the addition of two vectors amounts to the determination of a third vector having the same initial point as the first, its terminal point coinciding with the final point reached by carrying out the process of addition in the way indicated above.

When a and b have the same direction the numerical or scalar value of their sum is the same as that of the numerical measures of a and b: in all other cases the numerical or scalar value of the vector is algebraically less.

If from the initial point A in Fig. 1, the vector AB′ is drawn equal to BC, and then a vector B′C equal to the vector a is drawn from B′, its terminal point coincides with C, because the lines form a parallelogram and the addition of these two latter vectors gives the same vector diagonal c as in the first case, so that we may write,

which means that the sum of the two vectors is independent of the order in which they are taken. The analogy with the addition of ordinary algebraical quantities is to be noticed.

If now from C another vector CD is drawn (Fig. 2) (not in general in the plane of a and b), D will be reached by drawing the vector AD or e, which also is not in general in the plane of a and b.

FIG. 2.

But from the diagrammatic figure we see that, if CD = d,

This process may be continued with any number of vectors, and the result is, that the vector joining the initial point of the first of any number of vectors added together as in Fig. 2, to the terminal point of the last, is equivalent to the vector sum of them all.

Again, from the figure it is seen that

AD = AC + CD = AB + BD,

or, since         BD = BC + CD, (a + b) + d = a + (b + d).

This equation and equation (2) show that in vector addition the associative and commutative laws of algebra hold respectively.

From these laws extended to the case of any number of vectors, it may be proved that the sum of any number of vectors is independent of the order in which they are taken.

In the remainder of this section, however, this result is obtained by considering a particular case of a given number of vectors, and by observing that the method is applicable whatever their number may be.

The process of vector addition may be simplified sometimes by forming the sums of groups of the vectors to be added, and then adding these partial sums vectorially.

This is illustrated in Fig. 3, which is also diagrammatic, the vectors therein not being in one plane.

By the direct method just explained, the sum of the seven vectors, a1, a2, a3, ... a6, a7, is AH.

But we get the same result if we group, say a1, a2 and a3 together, and add their sum to the partial sum of a4 and a5, and then add the sum of the two groups to the sum of a6 and a7: for AD is equivalent to (a1 + a2 + a3), and DF to (a4 + a5): and FH to (a6 + a7), so that AF = AD + DF = (a1 + a2 + a3) + (a4 + a5), and in the same way,

FIG. 3.

This process of grouping, in the addition of vectors, is strictly analogous to the similar grouping into partial sums of ordinary algebraical quantities, which are to be added together, and is applicable to any number of vectors.

That the addition of n vectors can be made in any order without altering the result may be shown from the property that the order of any two vectors may be inverted as shown in Eqn. 2, in conjunction with the method of grouping explained above.

For if a1, a2, ... an−1 · an was the original order of the vectors, then any other order such as a2, an−1, ... an, a1, an−2 can be reached by successive interchanges of pairs of vectors. To take a concrete example, suppose the seven vectors of Fig. 3 are in the order there given, i.e. with suffixes 1, 2, 3, 4, 5, 6, 7, and we want to find their sum when their order is given by the suffixes 5, 6, 7, 2, 4, 1, 3.

Start with interchanging 4 and 5, then by the preceding and by Eqn. 2 the sum of the three groups of terms with suffixes (1, 2, 3) (4, 5) (6, 7) is equivalent to the sum of the groups (1, 2, 3) (5, 4) (6, 7), so that the sum of the vectors in the original order is the same as in the order 1, 2, 3, 5, 4, 6, 7.

By three more similar interchanges towards the left the suffix 5 comes to the first place, and the order is now 5, 1, 2, 3, 4, 6, 7, with an unaltered sum of the seven vectors. We then proceed in the same manner to get 6 in the second place, and then 7 in the third place, and so on until we reach the required order. Since the sum at each interchange of a pair of vectors is not altered, the sum of the seven vectors with the suffixes in the required order, i.e. 5, 6, 7, 2, 4, 1, 3, is the same as with the suffixes in the original order.

This method is applicable to the sum of any number of vectors.

[3].   Vectorial Signs and Subtraction of Vectors.

In the preceding Section the signs + and = were introduced, and the meaning of + was made clear: as regards the sign =, as already stated it signifies equivalence in magnitude and direction, so that the equation a = b signifies that the vector b is of the same numerical magnitude as a and is drawn in the same direction and in the same sense.

Thus if we have the equation,

a1 + a2 + ... + an = b1 + b2 + ... + bm,

the interpretation of it is that the vector representing the vector sum of the quantities on the left-hand side is equal in magnitude and direction to the vector representing the vector sum of the quantities on the right-hand side of the equation.

The sign – may be added with advantage to the two above-mentioned signs, and its meaning will become clear from the following considerations.

If we interpret the minus sign –, before a vector a as signifying that −a is a negative vector having the same numerical value as a (or +a), but as being drawn in the opposite direction, it follows that +a + (−a) = 0, and that −(−a) = +a.

From the equation a + b = c, we have

a + b − b = c − b, or a = c − b.

The vector b on the right-hand side may be considered as subtracted from c or the negative vector −b as added to c. From either point of view the result is the same, viz. a.

Thus if we introduce the idea of negative vectors the subtraction of one vector from another is equivalent to the addition to the first vector of a negative vector of equal numerical magnitude to the second, i.e. c − b = c + (−b), also c − (−b) = c + b.

Therefore in all cases the subtraction of one vector from another is equivalent to an operation of addition, and the term algebraical addition of vectors includes both addition and subtraction: in the same way that both the addition and subtraction of algebraical quantities are included in the term algebraical addition.

But the negative vector, as has been explained, differs from an ordinary algebraical negative quantity, for besides that its absolute magnitude is the same as in the corresponding positive vector, it must lie in the same direction as that of the positive vector, though drawn in an opposite sense.

FIG. 4.

To contrast the operations of addition and subtraction of two vectors, let AB = a, BC = b, BC′ = −b, then (Fig. 4) a + b = c, where c = AC, also a − b = d, where d = AC′. But AC′ = DB. Therefore if we form a parallelogram with a and b as adjacent sides, a + b is represented by the diagonal joining the initial point of a to the terminal point of b (the initial point of the latter coinciding with the terminal point of a), while the vector a − b is represented by the other diagonal.

[4].   Resolution of a vector into parts, one scalar, the other vectorial.

Every vector contains a certain number of scalar units of length, integral and fractional, and it has a certain definite direction. A vector of unit length may be indicated by attaching the subscript suffix 1 (in clarendon type).

Thus if we write the equation a = aa1, we mean that the vector a consists of "a" scalar units of length in the direction of the unit vector a1.

It is frequently convenient to use the notation | a | to denote the scalar part of the vector a, thus,

[5].   Resolution of a Vector into its Components.

From Sect. 2 it is evident that any vector may be considered as the sum of any number of other vectors or components. Thus in Fig. 3, Sect. 2, AD has a1, a2, a3 for its components, but it might also have arisen from the addition of any number of other components, subject to the condition that the terminal point of the last component vector coincides with the point D.

This property is utilised to split a vector into three components which do not lie in one plane.

Three components are necessary and sufficient to locate a vector in space of three dimensions. For two intersecting vectors define a plane, and a third vector not in their plane but passing through their common initial point or origin, enables all points in space to be referred to the three selected vectors.

The resolution of a vector drawn from the origin to any point in space into three components is effected by the following method.

In Fig. 5 let OA, OB, OC be three arbitrary vectors a, b, c, no two of which lie in the same plane with the third. Take the plane of the paper as that of AOB, then OC is not in this plane. Take any point P outside the plane of AOB, draw PE parallel to OC meeting the plane of AOB in E, and draw EF parallel to OB meeting OA in F. Then the vector OP which we may call p is the sum of OF, FE, and EP: that is,

FIG. 5.

p = OF + FE + EP.

Suppose

| OF |/| OA | = k1, | FE |/| OB | = k2, | EP |/| OC | = k3,

then         OF = k1OA, FE = k2OB, EP = k3OC,

so that        p = k1a + k2b + k3c,

or if            a = aa1, b = bb1 c = cc1, p = αa1 + βb1 + γc1,

where α, β, γ stand respectively for ak1, bk2, ck3, that is for the scalar parts of the components of OP.

If a, b, c each equal unity, then

in which expression a1, b1, c1, or OA, OB, OC are respectively taken as unit vectors in the directions of a, b, c.

The reader will notice that the absolute lengths of the three units a, b, c as drawn in the figure are different, and, indeed, they have been chosen arbitrarily.

In practice, however, they are usually taken of the same absolute length: this is especially the case, when instead of a, b, c the directional units i, j, k (shortly to be introduced) are employed.

Attention will now be drawn to an important vector property, and that is, that the resolution of a vector into three components drawn in given directions is unique, that is to say, that it can only be effected in one way.

For if the vector r could be resolved into two sets of components drawn respectively in the same directions, we should have the two relations,

where one at least of the scalar coefficients in the first equation differs from the corresponding coefficient in the second; combining the two equations, we have

r − r = 0 = (x1 − y1)a + (x2 − y2)b + (x3 − y3)c.

Let us assume that originally the first r was greater than the second, but gradually diminished in magnitude while keeping its direction, until its scalar magnitude became that of the second r.

Now if a vector diminishes in magnitude keeping its direction constant, the scalar parts of its three components diminish in the same ratio and vanish simultaneously with the vector; so that when a vector vanishes its scalar components vanish at the same time.

Therefore in the last equation we should have

(x1 − y1) = (x2 − y2) = (x3 − y3) = 0,

which is contrary to our assumption that one at least of the members of the last equation is different from zero.

Hence as stated above the resolution of a vector into three components drawn in given directions is unique.

It follows that if in an investigation, two vectors, r1 = x1a + x2b + x3c, and r2 = y1a, + y2b + y3c, are found to be equal, then necessarily

x1 = y1, x2 = y2, x3 = y3.

Applying this result to the case of the sum of n vectors v1, v2, v3, ... vn, if we resolve each into three components parallel to the vectors a, b, c, so that

v1 = a1a1 + b1b1 + c1c1,

v2 = a2a1 + b2b1 + c2c1, ..., vn = ana1 + bnb1 + cnc1,

and if r is the sum of the original n vectors, and is similarly resolved, so that r = ara1 + brb1 + crc1, then since

therefore

so that

The most usual way of resolving a vector is into three components at right angles to one another, their directions being assumed fixed.

Lines drawn parallel to these directions from a point constitute the usual X-, Y-, Z-axes with the origin at the point. Unit vectors in these directions would, according to Sect. 4, be expressed as x1, y1, z1, but in the case of rectangular axes it is usual to substitute i, j, k as unit vectors in the directions of the X-, Y-, Z-axes respectively. Thus if the vector v is the sum of three rectangular components, whose scalar values in the directions of the X-, Y- and Z-axes respectively are v1, v2, v3, we have the relation

When the vector is actually that connecting the origin to a point P, it is usual to denote the vector by r, and its scalar components by x, y, z, so that we have in this case

This particular vector, r, is frequently referred to as the position vector of the point P.

From ordinary analysis we know that the scalar rectangular components of a vector are equal to the scalar part of the vector multiplied by its direction cosines λ, μ, ν: thus, v = υλi + υμj + υνk.

If we are dealing with unit vector v1, | v1 | is unity, and we have

that is, the scalar components of a unit vector are its direction cosines.

[6].   On the determination of the direction of a vector.

It will be advisable at this point to get clear ideas of what constitutes the determination of the direction of a vector, for we have spoken of the drawing of the components of vectors parallel to fixed directions.

If I look at the moon, I say it is situated in a given direction at a given instant, and, perhaps unconsciously, I estimate its altitude above the horizon, and its bearing or azimuth referred to the cardinal points of the horizon. I thus refer the given direction of the moon to other directions, which for the purposes of reference I assume fixed, although I know that in reality they are not really so.

It follows, on reflection, that there is no meaning to be attached to such a term as absolute direction, for we always have to refer any given direction to other standard directions, which we take as fixed, though in fact they may be moving with respect to other so-called fixed directions with which we compare them.

For instance, we might take as our axes of reference at our point of observation on the earth, a line parallel to a normal to the plane of the ecliptic and two parallel directions to that plane, one parallel to the line passing through the equinoctial point at the vernal equinox drawn from the sun’s centre, the other at right angles to the latter.

Or we might take as axes at our point of observation a line parallel to the earth’s polar axis, and two perpendicular directions parallel to the plane of the equator, one, say, in the plane of the meridian through the place.

It is clear that we can refer any line drawn in any direction from the point of observation as origin, to either of these two sets of axes, although each set is in motion relative to the other.

The method of splitting up a vector into three rectangular components is a case in point, for we assume the three directions of the axes to be fixed for the purpose of reference to them, although as a whole they may be moving in any way whatever.

Again, while the direction and magnitude of a vector may remain unchanged, its components vary according to the directions of the axes to which we refer it.

One great advantage of vector methods is that we are enabled to obtain general results without reference to fixed axes of reference; the latter being introduced finally for the solution of any particular problem.

[7].   Systems of Coordinates.

In Sect. 5 it was stated that three components were required to determine the direction and magnitude of a vector in space of three dimensions.

It is not, however, necessary that the three coordinates should be referred to three rectilinear axes: for some of the rectilinear coordinates may be replaced by angles.

Thus a vector may be determined by means of a fixed plane passing through the initial point, the plane having a fixed line in it also passing through the initial point.

If now we drop a perpendicular from any point on the vector on to the plane, we can determine the angle φ made by the vector with the plane, and also the angle θ made by the trace of the vector in the plane with the fixed line in the plane. These two angles determine the direction of the vector; but to determine it completely we must know its scalar length.

The accompanying figure illustrates the above. OD is the vector; the plane of the paper is taken as the fixed plane, and OA is a fixed line in it; CB is a perpendicular from any point C in OD on to the plane of the paper, and the angle BOC = φ, and angle AOB = θ, between OB the trace of OC on the plane, and OA the fixed line in the plane.

FIG. 6.

There are other possible systems of coordinates: any that we may make use of will be described as they occur, and oblique coordinates will be dealt with in this section.

The property they all have in common is the possession of three independent quantities to which the vector can be referred. If the coordinates are straight lines, they must not be co-planar, for in that case it would not be possible to determine a vector outside the plane of reference.

In Sect. 5 the resolution of a vector into three components was explained. When the components are not at right angles, but make any angles with one another (although they must not all lie in one plane), the expression for the vector r is (Sect. 5, Eqn. 5)

x1, y1 and z1 being unit vectors drawn in the fixed directions of the axes of reference.

The coordinates of this system of resolution are known as oblique, and in many cases the adoption of this system is of advantage.

It is clear that the system of rectangular coordinates employed in the large majority of cases is a special case of oblique coordinates, the angle between each pair of axes being a right angle.

[8].   Vector equations of the Plane and the Straight Line.

By suitably varying x, y, z in the last equation, (9), namely, r = xx1 + yy1 + zz1, any vector whatever in space of three dimensions may be represented by it. Thus every point in space may be considered as being at the extremity of its corresponding position vector (Sect. 5) drawn from the origin.

In the particular case of one of the quantities x, y, or z being zero, say z, the equation becomes

and r may be made to express all vectors which lie in the plane of xy, by suitably varying x and y.

Since every point in this plane is at the extremity of its corresponding vector, the above equation may be termed the vector equation of the plane of xy.

Generally the equation

is the vector equation of the plane in which a and b lie when they are drawn from a common origin.

If both y and z vanish, the equation reduces to r = xx1, and it represents

Reviews for Problems and Worked Solutions in Vector Analysis

[peter3capell · Rating: 4 out of 5 stars]

This is a 2014 Dover reissue of a 1938 Macmillan and Company of the same name. It has stood the test of time well. The writing is clear and devoid of fog, and the book’s diagrams are unfailingly helpful in meeting the desideratum that they illustrate—quite literally illustrate—whatever it is that the vectors, or kinds of vectors, being so written about are are up to.

It is surprisingly easy to fluff one’s attempt to explain this stuff, even in an introductory work. For one, mathematics authors often eschew worked examples, and readers are left to ponder abstractions, where they are unable to even begin answer set questions. Not this author. This author holds the reader’s hand by employing worked examples.

In fact, I liked the book enough to purchase a copy. It’s a formatting thing.