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DIFFERENTIAL OPERATORS IN CURVILINEAR COORDINATES5 Cylindrical Coordinates Divergence ∇·A=
1 ∂ 1 ∂Aφ ∂Az (rAr ) + + r ∂r r ∂φ ∂z
Gradient (∇f )r =
∂f ; ∂r
(∇f )φ =
1 ∂f ; r ∂φ
(∇f )z =
Curl (∇ × A)r =
∂Aφ 1 ∂Az − r ∂φ ∂z
(∇ × A)φ =
∂Ar ∂Az − ∂z ∂r
(∇ × A)z =
1 ∂ 1 ∂Ar (rAφ ) − r ∂r r ∂φ
Laplacian 1 ∂ ∇ f = r ∂r 2
∂f r ∂r
+
1 ∂2f ∂2f + r 2 ∂φ2 ∂z 2
6
∂f ∂z
Laplacian of a vector 2
2
2
2
2
2
(∇ A)r = ∇ Ar −
(∇ A)φ = ∇ Aφ +
Ar 2 ∂Aφ − 2 2 r ∂φ r
Aφ 2 ∂Ar − 2 2 r ∂φ r
(∇ A)z = ∇ Az
Components of (A · ∇)B (A · ∇B)r = Ar
Aφ ∂Br Aφ B φ ∂Br ∂Br + + Az − ∂r r ∂φ ∂z r
(A · ∇B)φ = Ar
∂Bφ Aφ ∂Bφ ∂Bφ Aφ Br + + Az + ∂r r ∂φ ∂z r
(A · ∇B)z = Ar
Aφ ∂Bz ∂Bz ∂Bz + + Az ∂r r ∂φ ∂z
Divergence of a tensor (∇ · T )r =
Tφφ 1 ∂Tφr ∂Tzr 1 ∂ (rTrr ) + + − r ∂r r ∂φ ∂z r
(∇ · T )φ =
∂Tzφ Tφr 1 ∂ 1 ∂Tφφ (rTrφ ) + + + r ∂r r ∂φ ∂z r
(∇ · T )z =
1 ∂ 1 ∂Tφz ∂Tzz (rTrz ) + + r ∂r r ∂φ ∂z
7
Spherical Coordinates Divergence ∇·A=
∂Aφ 1 ∂ 1 1 ∂ 2 ) + ) + (r A (sin θA r θ r 2 ∂r r sin θ ∂θ r sin θ ∂φ
Gradient (∇f )r =
∂f ; ∂r
(∇f )θ =
1 ∂f ; r ∂θ
(∇f )φ =
1 ∂f r sin θ ∂φ
Curl (∇ × A)r =
∂ 1 ∂Aθ 1 (sin θAφ ) − r sin θ ∂θ r sin θ ∂φ
(∇ × A)θ =
∂Ar 1 ∂ 1 − (rAφ ) r sin θ ∂φ r ∂r
(∇ × A)φ =
1 ∂Ar 1 ∂ (rAθ ) − r ∂r r ∂θ
Laplacian 1 ∂ ∇ f = 2 r ∂r 2
∂f r ∂r 2
1 ∂ + 2 r sin θ ∂θ
∂f sin θ ∂θ
+
1 ∂2f r 2 sin2 θ ∂φ2
Laplacian of a vector 2
2
2
2
2
2
(∇ A)r = ∇ Ar −
(∇ A)θ = ∇ Aθ +
(∇ A)φ = ∇ Aφ −
∂Aφ 2Ar 2 ∂Aθ 2 cot θAθ 2 − − − r2 r 2 ∂θ r2 r 2 sin θ ∂φ
2 ∂Ar Aθ 2 cos θ ∂Aφ − − r 2 ∂θ r 2 sin2 θ r 2 sin2 θ ∂φ
Aφ 2 ∂Ar 2 cos θ ∂Aθ + + r 2 sin2 θ r 2 sin θ ∂φ r 2 sin2 θ ∂φ
8
Components of (A · ∇)B (A · ∇B)r = Ar
Aφ ∂Br Aθ B θ + Aφ B φ ∂Br Aθ ∂Br + + − ∂r r ∂θ r sin θ ∂φ r
(A · ∇B)θ = Ar
Aφ ∂Bθ cot θAφ Bφ ∂Bθ Aθ ∂Bθ Aθ Br + + + − ∂r r ∂θ r sin θ ∂φ r r
(A · ∇B)φ = Ar
∂Bφ Aφ ∂Bφ Aφ B r cot θAφ Bθ Aθ ∂Bφ + + + + ∂r r ∂θ r sin θ ∂φ r r
Divergence of a tensor (∇ · T )r =
1 ∂ 1 ∂ 2 (r Trr ) + (sin θTθr ) 2 r ∂r r sin θ ∂θ +
(∇ · T )θ =
1 ∂ 1 ∂ 2 (r Trθ ) + (sin θTθθ ) 2 r ∂r r sin θ ∂θ +
(∇ · T )φ =
∂Tφr Tθθ + Tφφ 1 − r sin θ ∂φ r
∂Tφθ cot θTφφ 1 Tθr + − r sin θ ∂φ r r
1 ∂ 1 ∂ 2 (r Trφ ) + (sin θTθφ ) 2 r ∂r r sin θ ∂θ +
