spherical coordinate system and its transformation to cartesian or rectangular and cylindrical coordinate system

In the spherical coordinate system, the three orthogonal surfaces are a sphere, a cone and a plane. P ( d, θ, φ} ) represents a point.

where   d represents radius of a sphere or distance from the origin to point P

θ represents the angle of elevation that is angle between z axis at the origin and to point.

φ is the azimuthal angle measured from X- axis.

Unit Vectors:

The unit vectors at a point are di, θj and φ k

where   di is perpendicular to spherical surface in increasing d direction.

θj is  perpendicular to the surface of cone towards increasing θ.

Φ is perpendicular to the shifted XZ plane in the increasing Φ direction.

The coordinate system is right handed.

Thus cross product of spherical unit vectors is: Continue reading “spherical coordinate system and its transformation to cartesian or rectangular and cylindrical coordinate system”

cylindrical coordinate system and its transformation to cartesian or rectangular coordinate system

A cylindrical coordinate system is used for cylindrical symmetrical problems (examples, cables, machine rotor etc.). This system does not have axes like cartesian coordinate system.

Any point P ( r, Θ ,z ) is represented by the intersection of three  mutually orthogonal surfaces.

where r the radius of a cylinder.

Θ is the azimuthal angle and

z is the constant plane is same as in cartesian coordinate system.

Representation of unit vectors in cylindrical coordinate system:

The three unit vectors can be represented as ri, Θj, zk. Continue reading “cylindrical coordinate system and its transformation to cartesian or rectangular coordinate system”