Basics of Fluid Mechanics, 2014a

A.1. VECTORS 577
The divergence of a vector equals
∇·N =
[
1 ∂
h 1 h 2 h 3 ∂q 1 (N 1h 2 h 3 )+
∂
∂q 2 (N 2h 3 h 1 )+
For general orthogonal coordinate system the curl is
[
ê 2 ∂
h 3 h 1 ∂q 3 (h 1 N 1 ) −
[
∇×N =
ê1 ∂
h 2 h 3 ∂q 2 (h 3 N 3 ) −
∂ ]
∂q 1 (h 3 N 3 )
The Laplacian of a scalar equals
∇ 2 φ =
[ (
1 ∂ h2 h 3
h 1 h 2 h 3 ∂q 1 h 1
]
∂
∂q 3 (N 3h 1 h 2 ) . (A.43)
]
+
∂
∂q 3 (h 2 N 2 )
[
+ ê3 ∂
h 1 h 2 ∂q 1 (h 2N 2 ) −
)
∂φ
∂q 1 + ∂ (
h3 h 1
∂q 2 h 2
∂
∂q 2 (h 1N 1 )
)
∂φ
∂q 2 + ∂ (
h1 h 2
∂q 3 h 3
] (A.44)
)]
∂φ
∂q 3
(A.45)
The following table showing the different values for selected orthogonal system.
Fig. -A.6. Parabolic coordinates by user WillowW using Blender.