Appendice A Operatori Scalari e Vettoriali - INFN

x = r sin θ cosϕ r = (x 2 + y 2 + z 2 ) 1/2zy = r sin θ sin ϕ → θ = arccos√z = r cosθϕ = arctg y/x(x 2 + y 2 + z 2 )∇ · ⃗u = 1 r 2 ∂∂r (r2 u r ) +1 ∂r sin θ ∂θ (sin θu θ) +1 ∂u ϕr sin θ ∂ϕ(C.7)[1 ∂∇ × ⃗u = ⃗i rr sin θ ∂θ (sin θ u ϕ) − ∂u ]θ∂ϕ[1 1 ∂u r⃗i θr sin θ ∂ϕ − ∂ ]∂r (r u ϕ) +[1 ∂⃗i ϕr ∂r (r u θ) − ∂u ]r∂θ+(C.8)∇ 2 f = 1r 2 ∂∂r= 1 r( )r2 ∂f∂r +1r 2 sin θ∂ 2∂r 2 (rf) +1r 2 sinθ∂∂θ∂∂θ( )sin θ∂f +1(sin θ∂f∂θ∂θ∂ 2 f∂ϕ 2r 2 sin 2 θ)+1 ∂ 2 fr 2 sin 2 θ ∂ϕ 2(C.9)∇ 2 ⃗u = ⃗i r[∇ 2 u r − 2r 2 u r −⃗i θ[∇ 2 u θ −⃗i ϕ[∇ 2 u ϕ −2r 2 sin 2 θ( ∂ (sin θ u )]∂θ θ) + ∂uϕ∂ϕ +]+u θr 2 sin 2 θ+ 2r 2 ∂u r∂θ− 2cos θr 2 sin 2 θuϕ 2 ∂u+ rr 2 sin 2 θ r 2 sinθ ∂ϕ∂u ϕ∂ϕ+ 2cos θr 2 sin 2 θ]∂u θ∂ϕ(C.10)In coordinate cilindriche (r, ϕ, z), con versori ⃗i r , ⃗i ϕ e ⃗i z :x = r cosϕ r = (x 2 + y 2 ) 1/2y = r sin ϕ → ϕ = arctg y/xz = z z = z∂f∇f = ⃗i r∂r + 1 ∂f⃗i ϕr ∂ϕ + ∂f⃗i z∂z∇ · ⃗u = 1 r∂∂r (r u r) + 1 ∂u ϕr ∂ϕ + ∂u z∂z(C.11)(C.12)