A problem which has cylindrical or spherical symmetry could be expressed and solved in the familiar cartesian coordinate system. However, the solution would fail to show the symmetry and in most cases would be needlessly complex. Therefore, throughout this, in addition to the coordinate system, the circular cylindrical and spherical coordinate systems will be used. all three will be examined together in order to illustrate the similarities and differences.
A point P is described by three coordinates, in Cartesian (x,y,z), in circular cylindrical
, and in spherical 
, as shown in figure below. The order of specifying the coordinatesis important and should be carefully followed. The angle 
is the same angle in both the cylindrical and spherical systems. But, in the order of the coordinates,
appears in second position in cylindrical, 
, and the third position is spherical, . The same symbol, r, is used in both cylindrical and spherical for two quite different things. In cylindrical coordinates r measures the distance from the z axis in a plane normal to the z axis, while in the spherical system r measures the distance from the origin to the point. It should be clear from the context of the problem which r is intended.
A point is also defiined by the intersection of three orthogonal surfaces, as shown in the figure 1-3 below. In cartesian coordinates, the surfaces are the infinite planes x= const., and z= const. In cylindrical coordinates, z= const. is the same infinite plane as in cartesian; 
= const. is a half plane with its edge along the z axis; r= const. is a right circular cylinder. These three surfaces are orthogonal and their intersection locates point P. in spherical coordinates, = const. is the same half plane as in cylindrical; r= const. is a sphere with its center at the origin;
= const. is a right circular cone whose axis is the z axis and whose vertex is at the origin. Note that is limited to the range
Figure 1-4 shows the three unit vectors at point P. In the cartesian system, the unit vectors have fixed directions, independent of the location of P. This is not true for the other two systems (except in case ax). Each unit vector is normal to its coordinate surface and is in the direction in which the coordinate increases. Notice that all three systems are right- handed:
= const. is a half plane with its edge along the z axis; r= const. is a right circular cylinder. These three surfaces are orthogonal and their intersection locates point P. in spherical coordinates, = const. is the same half plane as in cylindrical; r= const. is a sphere with its center at the origin;
= const. is a right circular cone whose axis is the z axis and whose vertex is at the origin. Note that is limited to the range
Figure 1-4 shows the three unit vectors at point P. In the cartesian system, the unit vectors have fixed directions, independent of the location of P. This is not true for the other two systems (except in case ax). Each unit vector is normal to its coordinate surface and is in the direction in which the coordinate increases. Notice that all three systems are right- handed:
The component forms of a vector in the three systems are
It should be noted that he components 
etc., are not generally constants but more often are functions of the coordinates in that particular system.
etc., are not generally constants but more often are functions of the coordinates in that particular system.
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