Appendix B 519
Since∇x 1 = ˆx 1 /h 1 ,the unit vector in thex 1 direction is
ˆx 1 =h 1 ∇x 1. (B.6)
Also, because the coordinates form a right-handed orthogonal system the unit vectors
are related by
ˆx 1 ×xˆ 2 = ˆx 3 ,ˆx 2 ׈x 3 = ˆx 1 ,ˆx 3 ׈x 1 = ˆx 2. (B.7)
Generalized divergence and curl
LetVbe an arbitrary vector
V=V 1 ˆx 1 +V 2 xˆ 2 +V 3 xˆ 3 (B.8)
and consider the divergence of the first term,
∇·(V 1 ˆx 1 ) = ∇·(V 1 xˆ 2 ×xˆ 3 )
= ∇·(V 1 h 2 ∇x 2 ×h 3 ∇x 3 )
= ∇(h 2 h 3 V 1 )·∇x 2 ×∇x 3
= ∇(h 2 h 3 V 1 )·
xˆ 2 ×xˆ 3
h 2 h 3
=
ˆx 1
h 2 h 3
·∇(h 2 h 3 V 1 )
=
1
h 1 h 2 h 3
∂
∂x 1
(h 2 h 3 V 1 ). (B.9)
Extending this to all three terms gives the general form for the divergence to be
∇·V=
1
h 1 h 2 h 3
(
∂
∂x 1
(h 2 h 3 V 1 )+
∂
∂x 2
(h 1 h 3 V 2 )+
∂
∂x 3
(h 1 h 2 V 3 )
)
. (B.10)
Now consider the curl of the first term of the arbitrary vector, namely
∇×(V 1 xˆ 1 ) = ∇×(V 1 h 1 ∇x 1 )
= ∇(V 1 h 1 )×∇x 1
=
1
h 1
∇(V 1 h 1 )׈x 1 (B.11)
and so
∇×V=
1
h 1
∇(V 1 h 1 )׈x 1 +
1
h 2
∇(V 2 h 2 )×xˆ 2 +
1
h 3
∇(V 3 h 3 )×xˆ 3. (B.12)
The component in the direction ofˆx 1 is
xˆ 1 ·∇×V =
1
h 2
∇(V 2 h 2 )׈x 2 ·xˆ 1 +
1
h 3
∇(V 3 h 3 )×xˆ 3 ·ˆx 1
=
1
h 2
∇(V 2 h 2 )·ˆx 2 ×xˆ 1 +
1
h 3
∇(V 3 h 3 )·xˆ 3 ׈x 1
=
1
h 3
xˆ 2 ·∇(V 3 h 3 )−
1
h 2
ˆx 3 ·∇(V 2 h 2 )
=
1
h 2 h 3
∂
∂x 2
(V 3 h 3 )−
1
h 2 h 3
∂
∂x 3
(V 2 h 2 ) (B.13)