1. Cylindrical & Spherical Coordinates
Dr. Raad Al-Bdeery

2. Cylindrical Coordinates

3. A cylindrical coordinate system is useful for solving problems having cylindrical symmetry. The location in free space is uniquely defined by three variables, r, , and z, as shown in the Figure in the next slide. The coordinate r is the radial distance in the x-y plane with the range (0≤ r <∞),  is the azimuth angle measured from the positive x-axis with the range (0≤  < 2π), and z is as previously defined in the Cartesian coordinate system with the range (-∞< z <∞)

4. Pointing in a direction tangential to the cylindrical surface
With pointing away from the origin along r
Pointing in a direction tangential to the cylindrical surface
And pointing along the vertical

5. Cylindrical Coordinates
●Coordinates r, , z
● Unit vectors (in directions of increasing coordinates)
● Position vector
● Vector components
A = Ar Aφ + Az
Components not constant, even if vector is constant  z
z=z1 plan 1
P=(r1, , z1)
R=r1 cylinder R1 z  r1 y x
 =
1 plane

6. In cylindrical coordinates, a vector is expressed as:
And like all unit vectors,
In cylindrical coordinates, a vector is expressed as: =
Where Ar, Aφ and Az are the components of A along the
The magnitude of A is obtained as explained before:

7. The position vector OP shown in the Figure has components along r and z only, Thus1 1
The dependence of R1 on is implicit since depends on

8. Spherical Coordinates

9. Spherical Coordinates
Coordinates R,  , 
Unit vectors (in directions of increasing coordinates)
Position vector
Vector components z P A  1 O 1 y x

10. In Spherical coordinates, the vector A shown in the Figure is
expressed as:
Where AR, A and Aφ are the components of A along the directions of 
The magnitude of A is obtained as follows: