Fundamentals of Plasma Physics

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)