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1.1 SCALARS AND VECTORS

The term scalar refers to a quantity whose value may be represented by a single (positive or negative) real number. The x, y, and z we use in basic algebra are scalars, and the quantities they represent are scalars. A vector quantity has both a magnitude1 and a direction in space. We are concerned with two- and three-dimensional spaces only, but vectors may be defined in n-dimensional space in more advanced applications. Force, velocity, acceleration, and a straight line from the positive to the negative terminal of a storage battery are examples of vectors. Each quantity is characterized by both a magnitude and a direction. Our work will mainly concern scalar and vector fields. A field (scalar or vector) may be defined mathematically as some function that connects an arbitrary origin to a general point in space. We usually associate some physical effect with a field, such as the force on a compass needle in the earth’s magnetic field, or the movement of smoke particles in the field defined by the vector velocity of air in some region of space. Note that the field concept invariably is related to a region. Some quantity is defined at every point in a region. Both scalar fields and vector fields exist. The temperature throughout the bowl of soup and the density at any point in the earth are examples of scalar fields. The gravitational and magnetic fields of the earth, the voltage gradient in a cable, and the temperature gradient in a soldering-iron tip are examples of vector fields. The value of a field varies in general with both position and time. In this book, as in most others using vector notation, vectors will be indicated by boldface type, for example, A. Scalars are printed in italic type, for example, A. When writing longhand, it is customary to draw a line or an arrow over a vector quantity to show its vector character.

<![if !supportLists]>1.2         <![endif]>VECTOR ALGEBRA

With the definition of vectors and vector fields now established, we may proceed to define the rules of vector arithmetic, vector algebra, and (later) vector calculus. Some of the rules will be similar to those of scalar algebra, some will differ slightly, and some will be entirely new.

To begin, the addition of vectors follows the parallelogram law. Figure 1.1 shows the sum of two vectors, A and B. It is easily seen that A+B = B+A, or that vector addition obeys the commutative law. Vector addition also obeys the associative law,

A + (B + C) = (A + B) + C

Note that when a vector is drawn as an arrow of finite length, its location is defined to be at the tail end of the arrow.

Coplanar vectors are vectors lying in a common plane, such as those shown in Figure 1.1. Both lie in the plane of the paper and may be added by expressing each vector in terms of “horizontal” and “vertical” components and then adding the corresponding components.

Vectors in three dimensions may likewise be added by expressing the vectors in terms of three components and adding the corresponding components. Examples of this process of addition will be given after vector components are discussed in Section 1.4.

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            The rule for the subtraction of vectors follows easily from that for addition, for we may always express A−B as A+(−B); the sign, or direction, of the second vector is reversed, and this vector is then added to the first by the rule for vector addition.

            Vectors may be multiplied by scalars. The magnitude of the vector changes, but its direction does not when the scalar is positive, although it reverses direction when multiplied by a negative scalar. Multiplication of a vector by a scalar also obeys the associative and distributive laws of algebra, leading to

(r + s)(A + B) = r (A + B) + s(A + B) = rA + rB + sA + sB

Division of a vector by a scalar is merely multiplication by the reciprocal of that scalar. The multiplication of a vector by a vector is discussed in Sections 1.6 and 1.7. Two vectors are said to be equal if their difference is zero, or A = B if A − B = 0.

            In our use of vector fields we shall always add and subtract vectors that are defined at the same point. For example, the total magnetic field about a small horseshoe magnet will be shown to be the sum of the fields produced by the earth and the permanent magnet; the total field at any point is the sum of the individual fields at that point.

            If we are not considering a vector field, we may add or subtract vectors that are not defined at the same point. For example, the sum of the gravitational force acting on a 150 lb f (pound-force) man at the North Pole and that acting on a 175 lb f person at the South Pole may be obtained by shifting each force vector to the South Pole before addition. The result is a force of 25 lb f directed toward the center of the earth at the South Pole; if we wanted to be difficult, we could just as well describe the force as 25 lb f directed away from the center of the earth (or “upward”) at the North Pole.2

1.3 THE RECTANGULAR

COORDINATE SYSTEM

To describe a vector accurately, some specific lengths, directions, angles, projections, or components must be given. There are three simple methods of doing this, and about eight or ten other methods that are useful in very special cases. We are going to use only the three simple methods, and the simplest of these is the rectangular, or rectangular cartesian, coordinate system.

In the rectangular coordinate system we set up three coordinate axes mutually at right angles to each other and call them the x, y, and z axes. It is customary to choose a right-handed coordinate system, in which a rotation (through the smaller angle) of the x axis into the y axis would cause a right-handed screw to progress in the direction of the z axis. If the right hand is used, then the thumb, forefinger, and middle finger may be identified, respectively, as the x, y, and z axes. Figure 1.2a shows a right-handed rectangular coordinate system.

A point is located by giving its x, y, and z coordinates. These are, respectively, the distances from the origin to the intersection of perpendicular lines dropped from the point to the x, y, and z axes. An alternative method of interpreting coordinate

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values, which must be used in all other coordinate systems, is to consider the point as being at the common intersection of three surfaces. These are the planes x =constant, y = constant, and z = constant, where the constants are the coordinate values of the point.

            Figure 1.2b shows points P and Q whose coordinates are (1, 2, 3) and (2,−2, 1), respectively. Point P is therefore located at the common point of intersection of the planes x = 1, y = 2, and z = 3, whereas point Q is located at the intersection of the planes x = 2, y = −2, and z = 1.

As we encounter other coordinate systems in Sections 1.8 and 1.9, we expect points to be located at the common intersection of three surfaces, not necessarily planes, but still mutually perpendicular at the point of intersection.

If we visualize three planes intersecting at the general point P, whose coordinates are x, y, and z, we may increase each coordinate value by a differential amount and obtain three slightly displaced planes intersecting at point P_, whose coordinates are x +dx, y +dy, and z +dz. The six planes define a rectangular parallelepiped whose volume is dv = dxdydz; the surfaces have differential areas dS of dxdy, dydz, an dzdx. Finally, the distance dL from P to P_ is the diagonal of the parallelepiped and has a length of _ (dx)2 + (dy)2 + (dz)2. The volume element is shown in Figure 1.2c; point P_ is indicated, but point P is located at the only invisible corner.

All this is familiar from trigonometry or solid geometry and as yet involves only scalar quantities.We will describe vectors in terms of a coordinate system in the next section.

1.4 VECTOR COMPONENTS

AND UNIT VECTORS

To describe a vector in the rectangular coordinate system, let us first consider a vector r extending outward from the origin. A logical way to identify this vector is by giving the three component vectors, lying along the three coordinate axes, whose vector sum must be the given vector. If the component vectors of the vector r are x, y, and z, then r = x+y+z. The component vectors are shown in Figure 1.3a. Instead of one vector, we now have three, but this is a step forward because the three vectors are of a very simple nature; each is always directed along one of the coordinate axes.

The component vectors have magnitudes that depend on the given vector (such as r), but they each have a known and constant direction. This suggests the use of unit vectors having unit magnitude by definition; these are parallel to the coordinate axes and they point in the direction of increasing coordinate values.We reserve the symbol a for a unit vector and identify its direction by an appropriate subscript. Thus ax , ay , and az are the unit vectors in the rectangular coordinate system.3 They are directed along the x, y, and z axes, respectively, as shown in Figure 1.3b.

If the component vector y happens to be two units in magnitude and directed toward increasing values of y, we should then write y = 2ay. A vector rP pointing<![if !vml]><![endif]>

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from the origin to point P(1, 2, 3) is written rP = ax + 2ay + 3az . The vector from P to Q may be obtained by applying the rule of vector addition. This rule shows that the vector from the origin to P plus the vector from P to Q is equal to the vector from the origin to Q. The desired vector from P(1, 2, 3) to Q(2,−2, 1) is therefore

RPQ = rQ − rP = (2 − 1)ax + (−2 − 2)ay + (1 − 3)az

= ax − 4ay − 2az

The vectors rP , rQ, and RPQ are shown in Figure 1.3c.

The last vector does not extend outward from the origin, as did the vector r we initially considered. However, we have already learned that vectors having the same magnitude and pointing in the same direction are equal, so we see that to help our visualization processes we are at liberty to slide any vector over to the origin before determining its component vectors. Parallelism must, of course, be maintained during the sliding process.

If we are discussing a force vector F, or indeed any vector other than a displacement-type vector such as r, the problem arises of providing suitable letters for the three component vectors. It would not do to call them x, y, and z, for these are displacements, or directed distances, and are measured in meters (abbreviated m) or some other unit of length. The problem is most often avoided by using component scalars, simply called components, Fx , Fy , and Fz . The components are the signed magnitudes of the component vectors.We may then write F = Fxax + Fyay + Fzaz . The component vectors are Fxax , Fyay , and Fzaz .

Any vector B then may be described by B = Bxax+Byay+Bzaz . The magnitude of B written |B| or simply B, is given by

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Each of the three coordinate systems we discuss will have its three fundamental and mutually perpendicular unit vectors that are used to resolve any vector into its component vectors. Unit vectors are not limited to this application. It is helpful to write a unit vector having a specified direction. This is easily done, for a unit vector in a given direction is merely a vector in that direction divided by its magnitude. A unit vector in the r direction is r/ _

x2 + y2 + z2, and a unit vector in the direction of the vector B is

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1.5 THE VECTOR FIELD

We have defined a vector field as a vector function of a position vector. In general, the magnitude and direction of the function will change as we move throughout the region, and the value of the vector function must be determined using the coordinate values of the point in question. Because we have considered only the rectangular coordinate system, we expect the vector to be a function of the variables x, y, and z.

If we again represent the position vector as r, then a vector field G can be expressed in functional notation as G(r); a scalar field T is written as T (r).

If we inspect the velocity of the water in the ocean in some region near the surface where tides and currents are important, we might decide to represent it by a velocity vector that is in any direction, even up or down. If the z axis is taken as upward, the x axis in a northerly direction, the y axis to the west, and the origin at the surface, we have a right-handed coordinate system and may write the velocity vector as v = vxax + vyay + vzaz, or v(r) = vx (r)ax + vy (r)ay + vz (r)az ; each of the components vx , vy , and vz may be a function of the three variables x, y, and z. If we are in some portion of the Gulf Stream where the water is moving only to the north, then vy and vz are zero. Further simplifying assumptions might be made if the velocity falls off with depth and changes very slowly as we move north, south, east, or west. A suitable expression could be v = 2ez/100ax. We have a velocity of 2 m/s (meters per second) at the surface and a velocity of 0.368 × 2, or 0.736 m/s, at a depth of 100 m (z = −100). The velocity continues to decrease with depth, while maintaining a constant direction.

1.6 THE DOT PRODUCT

We now consider the first of two types of vector multiplication. The second type will be discussed in the following section.

Given two vectors A and B, the dot product, or scalar product, is defined as the product of the magnitude of A, the magnitude of B, and the cosine of the smaller angle between them,

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The dot appears between the two vectors and should be made heavy for emphasis. The dot, or scalar, product is a scalar, as one of the names implies, and it obeys the commutative law,

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for the sign of the angle does not affect the cosine term. The expression A· B is read “A dot B.” Perhaps the most common application of the dot product is in mechanics, where a constant force F applied over a straight displacement L does an amount of work FL cos θ, which is more easily written F · L. We might anticipate one of the results of Chapter 4 by pointing out that if the force varies along the path, integration is necessary to find the total work, and the result becomes

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Another example might be taken from magnetic fields. The total flux _ crossing a surface of area S is given by BS if the magnetic flux density B is perpendicular to the surface and uniform over it. We define a vector surface S as having area for its magnitude and having a direction normal to the surface (avoiding for the moment the problem of which of the two possible normals to take). The flux crossing the surface is then B · S. This expression is valid for any direction of the uniform magnetic flux density. If the flux density is not constant over the surface, the total flux is _ = _ B · dS. Integrals of this general form appear in Chapter 3 when we study electric flux density.

Finding the angle between two vectors in three-dimensional space is often a job we would prefer to avoid, and for that reason the definition of the dot product is usually not used in its basic form.Amore helpful result is obtained by considering two vectors whose rectangular components are given, such as A = Axax + Ayay + Azaz and B = Bxax + Byay + Bzaz . The dot product also obeys the distributive law, and, therefore, A· B yields the sum of nine scalar terms, each involving the dot product of two unit vectors. Because the angle between two different unit vectors of the rectangular coordinate system is 90◦, we then have

ax · ay = ay · ax = ax · az = az · ax = ay · az = az · ay = 0

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The remaining three terms involve the dot product of a unit vector with itself, which is unity, giving finally

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which is an expression involving no angles.

A vector dotted with itself yields the magnitude squared, or

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and any unit vector dotted with itself is unity,

aA · aA = 1

One of the most important applications of the dot product is that of finding the component of a vector in a given direction. Referring to Figure 1.4a, we can obtain the component (scalar) of B in the direction specified by the unit vector a as

B · a = |B| |a| cos θBa = |B| cos θBa

The sign of the component is positive if 0 ≤ θBa ≤ 90◦ and negative whenever 90◦ ≤ θBa ≤ 180◦.

To obtain the component vector of B in the direction of a, we multiply the component (scalar) by a, as illustrated by Figure 1.4b. For example, the component of B in the direction of ax is B · ax = Bx , and the component vector is Bxax, or (B · ax )ax . Hence, the problem of finding the component of a vector in any direction becomes the problem of finding a unit vector in that direction, and that we can do.

The geometrical term projection is also used with the dot product. Thus, B · a is the projection of B in the a direction.

1.7 THE CROSS PRODUCT

Given two vectors A and B, we now define the cross product, or vector product, of A and B, written with a cross between the two vectors as A × B and read “A cross B.” The cross product A × B is a vector; the magnitude of A × B is equal to the product of the magnitudes of A, B, and the sine of the smaller angle between A and B; the direction of A×B is perpendicular to the plane containing A and B and is along one of the two possible perpendiculars which is in the direction of advance of a right-handed screw as A is turned into B. This direction is illustrated in Figure 1.5. Remember that either vector may be moved about at will, maintaining its direction constant, until the two vectors have a “common origin.” This determines the plane containing both. However, in most of our applications we will be concerned with vectors defined at the same point.

As an equation we can write

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where an additional statement, such as that given above, is required to explain the direction of the unit vector aN . The subscript stands for “normal.”

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Reversing the order of the vectors A and B results in a unit vector in the opposite direction, and we see that the cross product is not commutative, for B×A = −(A×B). If the definition of the cross product is applied to the unit vectors ax and ay, we find ax × ay = az , for each vector has unit magnitude, the two vectors are perpendicular, and the rotation of ax into ay indicates the positive z direction by the definition of a right-handed coordinate system. In a similar way, ay × az = ax and az × ax = ay . Note the alphabetic symmetry. As long as the three vectors ax , ay , and az are written in order (and assuming that ax follows az , like three elephants in a circle holding tails, so that we could also write ay , az , ax or az , ax , ay ), then the cross and equal sign may be placed in either of the two vacant spaces. As a matter of fact, it is now simpler to define a right-handed rectangular coordinate system by saying that ax × ay = az .

A simple example of the use of the cross product may be taken from geometry or trigonometry. To find the area of a parallelogram, the product of the lengths of two adjacent sides is multiplied by the sine of the angle between them. Using vector notation for the two sides, we then may express the (scalar) area as the magnitude of A × B, or |A × B|.

The cross product may be used to replace the right-hand rule familiar to all electrical engineers. Consider the force on a straight conductor of length L, where the direction assigned to L corresponds to the direction of the steady current I , and a uniform magnetic field of flux density B is present. Using vector notation, we may write the result neatly as F = IL × B.

The evaluation of a cross product by means of its definition turns out to be more work than the evaluation of the dot product from its definition, for not only must we find the angle between the vectors, but we must also find an expression for the unit vector aN . This work may be avoided by using rectangular components for the two vectors A and B and expanding the cross product as a sum of nine simpler cross products, each involving two unit vectors,

A × B = Ax Bxax ×ax + Ax Byax ×ay + Ax Bzax ×az

+ Ay Bxay ×ax + Ay Byay ×ay + Ay Bzay ×az

+ Az Bxaz ×ax + Az Byaz ×ay + Az Bzaz ×az

We have already found that ax × ay = az , ay × az = ax , and az × ax = ay . The three remaining terms are zero, for the cross product of any vector with itself is zero, since the included angle is zero. These results may be combined to give

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or written as a determinant in a more easily remembered form,

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Thus, if A = 2ax − 3ay + az and B = −4ax − 2ay + 5az , we have

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            = [(−3)(5) − (1(−2)]ax − [(2)(5) − (1)(−4)]ay + [(2)(−2) − (−3)(−4)]az

= −13ax − 14ay − 16az

1.8 OTHER COORDINATE SYSTEMS:

CIRCULAR CYLINDRICAL COORDINATES

The rectangular coordinate system is generally the one in which students prefer to work every problem. This often means a lot more work, because many problems possess a type of symmetry that pleads for a more logical treatment. It is easier to do now, once and for all, the work required to become familiar with cylindrical and spherical coordinates, instead of applying an equal or greater effort to every problem involving cylindrical or spherical symmetry later. With this in mind, we will take a careful and unhurried look at cylindrical and spherical coordinates.

The circular cylindrical coordinate system is the three-dimensional version of the polar coordinates of analytic geometry. In polar coordinates, a point is located in a plane by giving both its distance ρ from the origin and the angle φ between the line from the point to the origin and an arbitrary radial line, taken as φ = 0.4 In circular cylindrical coordinates, we also specify the distance z of the point from an arbitrary z = 0 reference plane that is perpendicular to the line ρ = 0. For simplicity, we usually refer to circular cylindrical coordinates simply as cylindrical coordinates. This will not cause any confusion in reading this book, but it is only fair to point out that there are such systems as elliptic cylindrical coordinates, hyperbolic cylindrical coordinates, parabolic cylindrical coordinates, and others.

We no longer set up three axes as with rectangular coordinates, but we must instead consider any point as the intersection of three mutually perpendicular surfaces. These surfaces are a circular cylinder (ρ = constant), a plane (φ = constant), and another plane (z = constant). This corresponds to the location of a point in a

rectangular coordinate system by the intersection of three planes (x = constant, y = constant, and z = constant). The three surfaces of circular cylindrical coordinates are shown in Figure 1.6a. Note that three such surfaces may be passed through any point, unless it lies on the z axis, in which case one plane suffices.

Three unit vectors must also be defined, but we may no longer direct them along the “coordinate axes,” for such axes exist only in rectangular coordinates. Instead, we take a broader view of the unit vectors in rectangular coordinates and realize that they are directed toward increasing coordinate values and are perpendicular to the surface on which that coordinate value is constant (i.e., the unit vector ax is normal to the plane x = constant and points toward larger values of x). In a corresponding way we may now define three unit vectors in cylindrical coordinates, aρ, aφ, and az .

The unit vector aρ at a point P(ρ1, φ1, z1) is directed radially outward, normal to the cylindrical surface ρ = ρ1. It lies in the planes φ = φ1 and z = z1. The unit vector aφ is normal to the plane φ = φ1, points in the direction of increasing φ, lies in the plane z = z1, and is tangent to the cylindrical surface ρ = ρ1. The unit vector az is the same as the unit vector az of the rectangular coordinate system. Figure 1.6b shows the three vectors in cylindrical coordinates.

In rectangular coordinates, the unit vectors are not functions of the coordinates. Two of the unit vectors in cylindrical coordinates, aρ and aφ, however, do vary with the coordinate φ, as their directions change. In integration or differentiation with respect to φ, then, aρ and aφ must not be treated as constants.

The unit vectors are again mutually perpendicular, for each is normal to one of the three mutually perpendicular surfaces, and we may define a right-handed cylindrical

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coordinate system as one in which aρ ×aφ =az, or (for those who have flexible fingers) as one in which the thumb, forefinger, and middle finger point in the direction of increasing ρ, φ, and z, respectively.

A differential volume element in cylindrical coordinates may be obtained by increasing ρ, φ, and z by the differential increments dρ, dφ, and dz. The two cylinders of radius ρ and ρ + dρ, the two radial planes at angles φ and φ + dφ, and the two “horizontal” planes at “elevations” z and z + dz now enclose a small volume, as shown in Figure 1.6c, having the shape of a truncated wedge. As the volume element becomes very small, its shape approaches that of a rectangular parallelepiped having sides of length dρ, ρdφ, and dz. Note that dρ and dz are dimensionally lengths, but dφ is not; ρdφ is the length. The surfaces have areas of ρ dρ dφ, dρ dz, and ρ dφ dz, and the volume becomes ρ dρ dφ dz.

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The variables of the rectangular and cylindrical coordinate systems are easily related to each other. Referring to Figure 1.7, we see that

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From the other viewpoint, we may express the cylindrical variables in terms of x, y, and z:

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We consider the variable ρ to be positive or zero, thus using only the positive sign for the radical in (11). The proper value of the angle φ is determined by inspecting the signs of x and y. Thus, if x = −3 and y = 4, we find that the point lies in the second quadrant so that ρ = 5 and φ = 126.9◦. For x = 3 and y = −4, we have φ = −53.1◦ or 306.9◦, whichever is more convenient.

Using (10) or (11), scalar functions given in one coordinate system are easily transformed into the other system.

A vector function in one coordinate system, however, requires two steps in order to transform it to another coordinate system, because a different set of component  vectors is generally required. That is, we may be given a rectangular vector

A = Axax + Ayay + Azaz

where each component is given as a function of x, y, and z, and we need a vector in cylindrical coordinates

A = Aρaρ + Aφaφ + Azaz

where each component is given as a function of ρ, φ, and z.

To find any desired component of a vector, we recall from the discussion of the dot product that a component in a desired direction may be obtained by taking the dot product of the vector and a unit vector in the desired direction. Hence,

Aρ = A· aρ and Aφ = A· aφ

Expanding these dot products, we have

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since az · aρ and az · aφ are zero.

In order to complete the transformation of the components, it is necessary to know the dot products ax · aρ, ay · aρ, ax · aφ, and ay · aφ. Applying the definition of the dot product, we see that since we are concerned with unit vectors, the result is merely the cosine of the angle between the two unit vectors in question. Referring to Figure 1.7 and thinking mightily, we identify the angle between ax and aρ as φ, and thus ax · aρ = cos φ, but the angle between ay and aρ is 90◦ − φ, and ay · aρ = cos (90◦ − φ) = sin φ. The remaining dot products of the unit vectors are found in a similar manner, and the results are tabulated as functions of φ in Table 1.1.

Transforming vectors from rectangular to cylindrical coordinates or vice versa is therefore accomplished by using (10) or (11) to change variables, and by using the dot products of the unit vectors given in Table 1.1 to change components. The two steps may be taken in either order.

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1.9 THE SPHERICAL COORDINATE SYSTEM

We have no two-dimensional coordinate system to help us understand the three dimensional spherical coordinate system, as we have for the circular cylindrical coordinate system. In certain respects we can draw on our knowledge of the latitude and- longitude system of locating a place on the surface of the earth, but usually we consider only points on the surface and not those below or above ground.

Let us start by building a spherical coordinate system on the three rectangular axes (Figure 1.8a). We first define the distance from the origin to any point as r . The surface r = constant is a sphere.

The second coordinate is an angle θ between the z axis and the line drawn from the origin to the point in question. The surface θ = constant is a cone, and the two surfaces, cone and sphere, are everywhere perpendicular along their intersection, which is a circle of radius r sin θ. The coordinate θ corresponds to latitude,

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except that latitude is measured from the equator and θ is measured from the “North Pole.”

The third coordinate φ is also an angle and is exactly the same as the angle φ of cylindrical coordinates. It is the angle between the x axis and the projection in the z = 0 plane of the line drawn from the origin to the point. It corresponds to the angle of longitude, but the angle φ increases to the “east.” The surface φ = constant is a plane passing through the θ = 0 line (or the z axis).

We again consider any point as the intersection of three mutually perpendicular surfaces—a sphere, a cone, and a plane—each oriented in the manner just described. The three surfaces are shown in Figure 1.8b.

Three unit vectors may again be defined at any point. Each unit vector is perpendicular to one of the three mutually perpendicular surfaces and oriented in that direction in which the coordinate increases. The unit vector ar is directed radially outward, normal to the sphere r = constant, and lies in the cone θ = constant and the plane φ = constant. The unit vector aθ is normal to the conical surface, lies in the plane, and is tangent to the sphere. It is directed along a line of “longitude” and points “south.” The third unit vector aφ is the same as in cylindrical coordinates, being normal to the plane and tangent to both the cone and the sphere. It is directed to the “east.”

The three unit vectors are shown in Figure 1.8c. They are, of course, mutually perpendicular, and a right-handed coordinate system is defined by causing ar ×aθ = aφ. Our system is right-handed, as an inspection of Figure 1.8c will show, on application of the definition of the cross product. The right-hand rule identifies the thumb, forefinger, and middle finger with the direction of increasing r , θ, and φ, respectively. (Note that the identification in cylindrical coordinates was with ρ, φ, and z, and in rectangular coordinates with x, y, and z.) A differential volume element may be constructed in spherical coordinates by increasing r , θ, and φ by dr, dθ, and dφ, as shown in Figure 1.8d. The distance between the two spherical surfaces of radius r and r + dr is dr; the distance between the two cones having generating angles of θ and θ + dθ is rdθ; and the distance between the two radial planes at angles φ and φ + dφ is found to be r sin θdφ, after a few moments of trigonometric thought. The surfaces have areas of r dr dθ, r sin θ dr dφ, and r 2 sin θ dθ dφ, and the volume is r 2 sin θ dr dθ dφ.

The transformation of scalars from the rectangular to the spherical coordinate system is easily made by using Figure 1.8a to relate the two sets of variables:

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The transformation in the reverse direction is achieved with the help of

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The radius variable r is nonnegative, and θ is restricted to the range from 0◦ to 180◦, inclusive. The angles are placed in the proper quadrants by inspecting the signs of x, y, and z.

The transformation of vectors requires us to determine the products of the unit vectors in rectangular and spherical coordinates. We work out these products from Figure 1.8c and a pinch of trigonometry. Because the dot product of any spherical unit vector with any rectangular unit vector is the component of the spherical

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vector in the direction of the rectangular vector, the dot products with az are found to be

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The dot products involving ax and ay require first the projection of the spherical unit vector on the xy plane and then the projection onto the desired axis. For example, ar · ax is obtained by projecting ar onto the xy plane, giving sin θ, and then projecting sin θ on the x axis, which yields sin θ cos φ. The other dot products are found in a like manner, and all are shown in Table 1.2.