Mathematical Methods for Physics and Engineering - Matematica.NET

10.9 CYLINDRICAL AND SPHERICAL POLAR COORDINATES∇Φ = ∂Φ∂r êr + 1 ∂Φr ∂θ êθ + 1 ∂Φr sin θ ∂φ êφ∇ · a =∇×a =∇ 2 Φ =1 ∂r 2 ∂r (r2 a r )+ 1 ∂r sin θ ∂θ (sin θa θ)+ 1 ∂a φr sin θ ∂φ∣ ê r rê θ r sin θ ê φ ∣∣∣∣∣∣∣1∂ ∂ ∂r 2 sin θ∂r ∂θ ∂φ∣ a r ra θ r sin θa φ(1 ∂r 2 ∂Φ )+ 1 (∂sin θ ∂Φ )+r 2 ∂r ∂r r 2 sin θ ∂θ ∂θ1r 2 sin 2 θ∂ 2 Φ∂φ 2Table 10.3 Vector operators in spherical polar coordinates; Φ is a scalar fieldand a is a vector field.zdφdrθrdθrdθr sin θdφφdφyr sin θr sin θdφxFigure 10.10 The element of volume in spherical polar coordinates is givenby r 2 sin θdrdθdφ.we can rewrite the first term on the RHS as follows:(1 ∂r 2 r 2 ∂Φ )= 1 ∂ 2∂r ∂r r ∂r 2 (rΦ),which can often be useful in shortening calculations.363