Differential analysis - Spherical Co-ordinate System - Field Theory.
Always remember that for a small change in angle the radius remains constant. Radius of a cylinder (ρ) is constant for a small change in φ . Similarly the radius of a sphere (r) is constant for a small change in θ. Therefore ρ always associates dφ , and r always associates dθ. Differential Length (dl) : - dl = dra r + r dθa θ + ρ dφ a φ We know that ρ = rsinθ Therefore dl = dra r + r dθa θ + r sinθdφ a φ Differential length for a surface is given as: - dl = dra r + r dθa θ ---( For r - θ plane or φ constant plane) - dl = r dθa θ + r sinθdφa φ ---( For θ - φ plane or r constant plane) - dl = dra r + r sinθdφ a φ ---( For r - φ plane or θ constant plane) Differential length for a line parallel to r, θ and φ axis are respectively given as: - dl = dra r ---(For a line parallel to r axis) - dl = r dθa θ ---( For a line parallel to θ axis) ∫ dl = ∫ o π rdθ = r (π - 0) = π r This answer describes the circumference of a semi circle. ...