Mathematics — Coordinate Systems

 

[Rectangular Coordinate System]

Cartesian coordinate system: another name

Cartesian coordinate system in one dimension: a point x on a straight line.

             rec_coordi_1d

Cartesian coordinate system in two dimensions: a point (x, y) on the xy-plane.

             rec_coordi_2d

Cartesian coordinate system in three dimensions:

            rec_coord_3d

Base vectors:

    : in the direction of increasing x with y and z held constant.

    : in the direction of increasing y with z and x held constant.

    : in the direction of increasing z with x and y held constant.

They are all perpendicular to each other:

   

Base vectors  are constant once they are defined.

The right-hand rule:

Coordinate of a point:

Position vector: a vector from (0, 0, 0) to (x, y, z).

Incremental length:

Incremental area:

Incremental volume:

                

[Cylindrical Coordinate System]

Cylindrical coordinate system in two dimensions:

                  polar_coordi

Base vectors:

Polar-to-rectangular base vector conversion:

Since  is on the xy-plane, it can be represented by a combination of  and :

Angle between  and :   ¡æ 

Angle between  and :   ¡æ 

    Thus,

Angle between  and :   ¡æ 

Angle between  and :   ¡æ 

    Thus,

In summary,

    ,

Conversely we have

    ,

Coordinate of a point:

    : distance from (0, 0) to the point

    : angle measured in counterclockwise direction from the positive x axis to the line segment from (0, 0) to

Polar-to-rectangular coordinate conversion:

   

   

Position vector:

Cylindrical coordinate system in three dimensions:

              

Base vectors: . They are all perpendicular to each other. Base vectors  change directions as the point moves.

The right-hand rule:  

Cylindrical-to-rectangular base vector conversion:

    , .

    , .

Coordinate of a point:

Cylindrical-to-rectangular coordinate conversion:

   

   

Position vector:

Incremental length:

Incremental area:

Incremental volume:  

             

 

[Spherical Coordinate System]

   

 

Base vectors: . They are all perpendicular to each other. Base vectors change directions as the point   moves.

The right-hand rule:

Spherical-to-rectangular base vector conversion:

Angle between  and :   ¡æ 

Angle between  and xy-plane:   ¡æ 

    Thus,

Angle between  and :   ¡æ 

Angle between  and xy-plane:   ¡æ 

    Thus,  

Angle between  and :   ¡æ 

Angle between  and :   ¡æ 

    Thus,

           spherical_and_cylindrical_coordinate_system.gif

In summary,

   

   

   

Conversely we can obtain:

   

     

   

Coordinate of a point:

Spherical-to-rectangular coordinate conversion:

   

   

Position vector:

Incremental length:

Incremental area:

Incremental volume: