System which uses a pair of Numbers or coordinates to define the Position of a Point on a plane, is called as coordinate system.
One of the type of coordinate system is spherical coordinate system which can be defined as a system which describes the position of point. This point may be located anywhere in Coordinate Plane. Normally spherical coordinate plane is expressed by three quantities.
A) Radial distance – This is also called radius or radial coordinate. Radial distance is the distance of point under measurement from origin.
B) Polar angle is an angle measured from zenith direction. Zenith direction is a fixed direction.
C) Azimuth angle - This angle is also called as inclination angle. This is the angle which usually passes through origin of spherical coordinate system. Origin is the Intersection of x – axes and y – axes.
ρ, θ, and φ are Spherical Coordinates. Here 'ρ' represents radial coordinate, 'θ' represents polar angle, 'φ' is represents azimuthal angle.
Volume element of spherical coordinate plane can be defined as:
For any volume 'P'
dP = ρ (t1, t2, t3) dt1 dt2 dt3,
Where t1, t2, t3….. are coordinates. For any Set 'A', volume is calculated by Integration as shown below:
Volume (A) = ∫ρ (t1, t2, t3) dt1 dt2 dt3
Lets take an example to understand the gradient in spherical coordinates
A Scalar field in terms of spherical coordinates is expressed below
B = B (ρ, θ, φ),
We know that gradient is written by using grad as prefix before a scalar quantity.
gradient in spherical coordinates will be written as:
Grad B = (∂ B / ∂ ρ) eρ^ + 1 / ρ (∂ B/ ∂ θ) eθ^+1 / ρ sin θ ( ∂ B / ∂ φ ) eφ^.