2)  Special Cases of the Orthogonal Coordinate Systems

The most frequently used coordinate systems are cartesian, circular cylindricaland spherical systems.

a) Cartesian coordinate system

The position of a point A in the cartesian coordinate system is given by the intersection of the three plane coordinate surfaces, see Fig. 2.1.



Fig.2.1, To the definition of the cartesian coordinate system

In this system the coordinates and basic unit vectors are denoted as

      
       (2.1)
      

Length, surface and volume elements are given by

(2.2)
(2.3)
(2.4)
(2.5)

The elementary volume element has a form of a parallelepiped, see Fig. 2.2



Fig.2.2, To the definition of the elementary volume in the carthesian coordinate system


b) Circular cylindrical coordinate system

The position of a point A in this system is given by the intersection of three coordinate surfaces: the cylindrical one - r = const., the plane surface - y = const. and the plane surface z = const., see Fig.2.3.



Fig.2.3, To the definition of the circular cylindrical coordinate system

The relations between the cylindrical and cartesian coordinates r, y, z and x, y, z can be obtained using geometry of Fig.2.3. One can obtain the following formulae



(2.6)

Basic units vectors of the local coordinate system are denoted here by



(2.7)

According to the Fig.2.4 the length, surface and volume elements are given by


(2.8)
, ,
(2.9)

(2.10)
(2.11)




Fig.2.4, To the definition of the elementary length, surface and volume elements in the circular cylindrical coordinate system

c) Spherical coordinate system

The position of a point A in this system is given by the intersection of the plain surface y = const., the conical surface J = const. and the spherical surface r = const., see Fig.2.5.



Fig.2.5, To the definition of the spherical coordinate system

Also in this case we can write analogical relations between (r, J, y) and (x, y, z) as it was in the case of cylindrical system.



(2.12)


(2.13)
, ,
(2.14)
, , (2.15)

For the elements the following relations hold


(2.16)

(2.17)
(2.18)

Elementary volume in spherical system has nearly the form of parallelepiped, see Fig.2.6.



Fig.2.6, To the definition of elementary volume in the spherical coordinate system