Cylindrical & Spherical Coordinates Dr. Raad Al-Bdeery

Cylindrical Coordinates

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 <∞)

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

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

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:

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 .

Spherical Coordinates

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

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: