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.
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:
di x θj = φk
θj x φk= di
φk x di = θj
The relation between the variables of cylindrical and spherical coordinates:
The relation between the variables of cylindrical and spherical coordinates and vice versa is given by:
r = dsinθ
Θ = φ
z = dcosθ
d = square root of (r2 + z2)
θ = tangent inverse (d/z)
The relation between the variables of cartesian or rectangular coordinates and spherical coordinates and vice versa:
The relation between the variables of cartesian or rectangular coordinates and spherical coordinates and vice versa is given by:
x = dsinθcosφ
y = rsinθsinφ
d = square root of (x2 + y2 + z2)
θ = cosine inverse [z/ square root of (x2 + y2 + z2)]
φ = tangent inverse (y/x)
Reference: This article is referred from my authored book “concepts of electromagnetic field theory” having ISBN 978-81-272-5245-8. Try to make the figures for spherical coordinate system. In case of any doubt in this article or any other EMFT or physics related article, kindly post in the comment section.