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 cartesian axes (Fig. 1.8a). We first define the distance from the origin to any point as r.

The surfacerˆconstant is a sphere.

FIGURE 1.8

(a) The three spherical coordinates. (b) The three mutually perpendicular surfaces of the spherical coordi- nate system. (c) The three unit vectors of spherical coordinates:araˆa. (d) The differential volume element in the spherical coordinate system.

The second coordinate is an angle between thezaxis and the line drawn from the origin to the point in question. The surfaceˆconstant is a cone, and the two surfaces, cone and sphere, are everywhere perpendicular along their intersection, which is a circle of radiusrsin. The coordinate corresponds to latitude, except that latitude is measured from the equator and is measured from the ``North Pole.''

The third coordinateis also an angle and is exactly the same as the angle of cylindrical coordinates. It is the angle between thexaxis and the projection in thezˆ0 plane of the line drawn from the origin to the point. It corresponds to the angle of longitude, but the angleincreases to the ``east.'' The surfaceˆ constant is a plane passing through theˆ0 line (or the zaxis).

We should again consider any point as the intersection of three mutually perpendicular surfacesÐa sphere, a cone, and a planeÐeach oriented in the manner described above. The three surfaces are shown in Fig. 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 rˆ constant, and lies in the cone ˆ constant and the planeˆconstant. The unit vectora 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 a is the same as in cylindrical coordinates, being normal to the plane and tangent to both the cone and sphere. It is directed to the ``east.''

The three unit vectors are shown in Fig. 1:8c. They are, of course, mutually perpendicular, and a right-handed coordinate system is defined by causing ara ˆa. Our system is right-handed, as an inspection of Fig. 1:8c will show, on application of the definition of the cross product. The right-hand rule serves to identify the thumb, forefinger, and middle finger with the direction of increasingr,, and, respectively. (Note that the identification in cylindrical coordinates was with; , andz, and in cartesian coordinates withx;y, andz). A differential volume element may be constructed in spherical coordinates by increasing r, , and bydr, d, and d, as shown in Fig. 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 and ‡d is rd; and the distance between the two radial planes at angles and ‡d is found to bersind, after a fewmoments of trigonometric thought. The surfaces have areas ofr dr d,rsindr d, andr2sindd, and the volume isr2sindr dd:

The transformation of scalars from the cartesian to the spherical coordinate system is easily made by using Fig. 1:8ato relate the two sets of variables:

xˆrsincos yˆrsinsin zˆrcos

…15†

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

x2‡y2‡z2

p …r0†

ˆcos 1 z



x2‡y2‡z2

p …081808†

ˆtan 1y x

…16†

The radius variableris nonnegative, and is restricted to the range from 08 to 1808, inclusive. The angles are placed in the proper quadrants by inspecting the signs ofx;y, andz.

The transformation of vectors requires the determination of the products of the unit vectors in cartesian and spherical coordinates. We work out these prod- ucts from Fig. 1:8cand a pinch of trigonometry. Since the dot product of any spherical unit vector with any cartesian unit vector is the component of the spherical vector in the direction of the cartesian vector, the dot products with az are found to be

azarˆcos azaˆ sin azaˆ0

The dot products involving ax and ay require first the projection of the spherical unit vector on the xyplane and then the projection onto the desired axis. For example,arax is obtained by projectingar onto the xyplane, giving sin, and then projecting sin on the xaxis, which yields sincos. The other dot products are found in a like manner, and all are shown in Table 1.2.

hExample 1.4

We illustrate this transformation procedure by transforming the vector field Gˆ …xz=y†axinto spherical components and variables.

Solution. We find the three spherical components by dottingGwith the appropriate unit vectors, and we change variables during the procedure:

TABLE 1.2

Dot products of unit vectors in spherical and cartesian coordinate systems

ar a a

ax sincos coscos sin

ay sinsin cossin cos

az cos sin 0

GrˆGarˆxz

y axarˆxz

ysincos

ˆrsincoscos2 sin GˆGaˆxz

yaxaˆxz

y coscos

ˆrcos2cos2 sin GˆGaˆxz

yaxaˆxz y… sin†

ˆ rcoscos Collecting these results, we have

Gˆrcoscos…sincotar‡coscota a†

Appendix A describes the general curvilinear coordinate system of which the cartesian, circular cylindrical, and spherical coordinate systems are special cases. The first section of this appendix could well be scanned now.

\ D1.7. Given the two points,C… 3;2;1† andD…rˆ5; ˆ208,ˆ 708†, find: (a) the spherical coordinates ofC; (b) the cartesian coordinates ofD; (c) the distance fromCto D:

Ans.C…rˆ3:74,ˆ74:58,ˆ146:38†;D…xˆ0:585;yˆ 1:607;zˆ4:70†; 6.29

\ D1.8. Transform the following vectors to spherical coordinates at the points given: (a) 10ax at P…xˆ 3, yˆ2, zˆ4); (b) 10ay at Q…ˆ5; ˆ308, zˆ4); (c) 10az at M…rˆ4; ˆ1108,ˆ1208).

Ans. 5:57ar 6:18a 5:55a; 3:90ar‡3:12a‡8:66a; 3:42ar 9:40a

SUGGESTED REFERENCES

1. Grossman, S. I.: ``Calculus,'' 3d ed., Academic Press and Harcourt Brace Jovanovich, Publishers, Orlando, 1984. Vector algebra and cylindrical and spherical coordinates appear in chap. 17, and vector calculus is introduced in chap. 20.

2. Spiegel, M. R.: ``Vector Analysis,'' Schaum Outline Series, McGraw-Hill Book Company, NewYork, 1959. A large number of examples and problems with answers are provided in this concise, inexpensive member of an outline series.

3. Swokowski, E. W.: ``Calculus with Analytic Geometry,'' 3d ed., Prindle, Weber, & Schmidt, Boston, 1984. Vector algebra and the cylindrical and spherical coordinate systems are discussed in chap. 14, and vector calculus appears in chap. 18.

4. Thomas, G. B., Jr., and R. L. Finney: ``Calculus and Analytic Geometry,'' 6th ed., Addison-Wesley Publishing Company, Reading, Mass., 1984. Vector algebra and the three coordinate systems we use are discussed in chap. 13.

Other vector operations are discussed in chaps. 15 and 17.

PROBLEMS

1.1 Given the vectors Mˆ 10ax‡4ay 8az and Nˆ8ax‡7ay 2az, find: (a) a unit vector in the direction of M‡2N; (b) the magnitude of 5ax‡N 3M; (c)jMjj2Nj…M‡N†:

1.2 Given three points,A…4;3;2†,B… 2;0;5†, andC…7; 2;1†: (a) specify the vector A extending from the origin to point A; (b) give a unit vector extending from the origin toward the midpoint of lineAB; (c) calculate the length of the perimeter of triangleABC:

1.3 The vector from the origin to pointAis given as 6ax 2ay 4az, and the unit vector directed from the origin toward pointBis 23; 23;13

. If points A andB are 10 units apart, find the coordinates of pointB:

1.4 Given pointsA…8; 5;4†andB… 2;3;2†, find: (a) the distance fromAto B; (b) a unit vector directed fromAtowardsB; (c) a unit vector directed from the origin toward the midpoint of the lineAB; (d) the coordinates of the point on the line connectingAtoBat which the line intersects the plane zˆ3:

1.5 A vector field is specified asGˆ24xyax‡12…x2‡2†ay‡18z2az. Given two points,P…1;2; 1†andQ… 2;1;3†, find: (a)GatP; (b) a unit vector in the direction ofGatQ; (c) a unit vector directed fromQtowardP; (d) the equation of the surface on whichjGj ˆ60:

1.6 For theGfield given in Prob. 1.5 above, make sketches ofGxGy,Gzand jGjalong the line yˆ1,zˆ1, for 0x2:

1.7 Given the vector field Eˆ4zy2cos 2xax‡2zysin 2xay‡y2sin 2xaz, find, for the region jxj, jyj, and jzj<2: (a) the surfaces on which Eyˆ0; (b) the region in whichEy ˆEz; (c) the region for whichEˆ0:

1.8 Two vector fields are Fˆ 10ax‡20x…y 1†ay and Gˆ2x2yax

4ay‡zaz. For the pointP…2;3; 4†, find: (a)jFj; (b)jGj; (c) a unit vector in the direction ofF G; (d) a unit vector in the direction ofF‡G:

1.9 A field is given asGˆ 25

x2‡y2…xax‡yay†. Find: (a) a unit vector in the direction ofGatP…3;4; 2†; (b) the angle betweenGandaxatP; (c) the value of the double integral„4

xˆ0

„2

zˆ0Gdx dzay on the planeyˆ7:

1.10 Use the definition of the dot product to find the interior angles atAand Bof the triangle defined by the three points:A…1;3; 2†,B… 2;4;5†, and C…0; 2;1†:

1.11 Given the pointsM…0:1; 0:2; 0:1†,N… 0:2;0:1;0:3†, andP…0:4;0;0:1†, find: (a) the vector RMN; (b) the dot product RMNRMP; (c) the scalar projection of RMN onRMP; (d) the angle between RMN andRMP:

1.12 Given points A…10;12; 6†, B…16;8; 2†, C…8;1;4†, and D… 2; 5;8†, determine: (a) the vector projection ofRAB‡RBC onRAD; (b) the vector projection ofRAB‡RBC onRDC; (c) the angle between RDA andRDC: 1.13 (a) Find the vector component ofFˆ10ax 6ay‡5azthat is parallel to

Gˆ0:1ax‡0:2ay‡0:3az. (b) Find the vector component of F that is perpendicular toG. (c) Find the vector component ofGthat is perpen- dicular toF:

1.14 The three vertices of a regular tetrahedron are located at O…0;0;0†, A…0;1;0†,B…0:5 

p3

;0:5;0†, andC… 

p3

=6;0:5; 

p2=3

†. (a) Find a unit vec- tor perpendicular (outward) to faceABC; (b) Find the area of faceABC:

1.15 Three vectors extending from the origin are given as r1ˆ 7ax‡3ay 2az, r2 ˆ 2ax‡7ay 3az, and r3 ˆ2ax 2ay‡3az. Find: (a) a unit vector perpendicular to bothr1 andr2; (b) a unit vector perpendicular to the vectors r1 r2 and r2 r3; (c) the area of the tri- angle defined by r1 and r2; (d) the area of the triangle defined by the heads ofr1;r2, andr3:

1.16 Describe the surface defined by the equation: (a) rax ˆ2, where rˆxax‡yay‡zaz; (b)jraxj ˆ2:

1.17 Point A… 4;2;5† and the two vectors, RAM ˆ20ax‡18ay 10az and RAN ˆ 10ax‡8ay‡15az, define a triangle. (a) Find a unit vector per- pendicular to the triangle. (b) Find a unit vector in the plane of the triangle and perpendicular to RAN. (c) Find a unit vector in the plane of the triangle that bisects the interior angle at A:

1.18 Given pointsA…ˆ5,ˆ708,zˆ 3†andB…ˆ2, ˆ 308, zˆ1†, find: (a) a unit vector in cartesian coordinates atAdirected towardB; (b) a unit vector in cylindrical coordinates atAdirected towardB; (c) a unit vector in cylindrical coordinates atB directed towardA:

1.19 (a) Express the vector field Dˆ …x2‡y2† 1…xax‡yay† in cylindrical components and cylindrical variables. (b) Evaluate D at the point whereˆ2,'ˆ0:2(rad), andzˆ5. Express the result in both cylind- rical and cartesian components.

1.20 Express in cartesian components: (a) the vector at A…ˆ4, ˆ408, zˆ 2) that extends to B…ˆ5, ˆ 1108, zˆ2); (b) a unit vector atBdirected towardA; (c) a unit vector atBdirected toward the origin.

1.21 Express in cylindrical components: (a) the vector from C…3;2; 7† to D… 1; 4;2†; (b) a unit vector atDdirected towardC; (c) a unit vector atD directed toward the origin.

1.22 A field is given in cylindrical coordinates as Fˆ 40

2‡1‡3…cos‡

sin†

a‡3…cos sin†a 2az. Prepare simple sketches of jFj: (a) vswith ˆ3; (b) vs with ˆ0; (c) vs with ˆ458:

1.23 The surfacesˆ3 and 5,ˆ1008and 1308, andzˆ3 and 4.5 identify a closed surface. (a) Find the volume enclosed. (b) Find the total area of the enclosing surface. (c) Find the total length of the twelve edges of the

surface. (d) Find the length of the longest straight line that lies entirely within the volume.

1.24 At point P… 3; 4;5†, express that vector that extends from P to Q…2;0; 1† in: (a) rectangular coordinates; (b) cylindrical coordinates;

(c) spherical coordinates. (d) Showthat each of these vectors has the same magnitude.

1.25 Let Eˆ 1

r2 cosar‡sin sina

. Given point P…rˆ0:8, ˆ308, ˆ458†, determine: (a) E at P; (b) jEj at P; (c) a unit vector in the direction ofE atP:

1.26 (a) Determine an expression for ay in spherical coordinates at P…rˆ4, ˆ0:2, ˆ0:8†. (b) Expressar in cartesian components at P:

1.27 The surfacesrˆ2 and 4,ˆ308and 508, andˆ208and 608identify a closed surface. (a) Find the enclosed volume. (b) Find the total area of the enclosing surface. (c) Find the total length of the twelve edges of the surface. (d) Find the length of the longest straight line that lies entirely within the volume.

1.28 (a) Determine the cartesian components of the vector from A…rˆ5, ˆ1108,ˆ2008†toB…rˆ7,ˆ308,ˆ708†. (b) Find the spherical components of the vector at P…2; 3;4† extending to Q… 3;2;5†. (c) If Dˆ5ar 3a‡4a, findDa at M…1;2;3†:

1.29 Express the unit vectoraxin spherical components at the point: (a)rˆ2, ˆ1 rad,ˆ0:8 rad; (b)xˆ3,yˆ2,zˆ 1; (c)ˆ2:5,ˆ0:7 rad, zˆ1:5:

1.30 Given A…rˆ20, ˆ308, ˆ458) and B…rˆ30, ˆ1158, ˆ1608), find: (a)jRABj; (b)jRACj, given C…rˆ20,ˆ908,ˆ458); (c) the dis- tance from AtoC on a great circle path.

2