Begin2.DVI

The first term in equation (8.94) can be expanded, and

∇(Fuˆeu) = ∇(Fu)·ˆeu+Fu∇(ˆeu)

=

1

h 1

∂F u

∂u +Fu∇[h^2 h^3 (∇v)×(∇w)] (See eqs. (8.89) and (8.93))

=^1

h 1

∂F u

∂u

+Fu{∇ (h 2 h 3 )·[(∇v)×(∇w)] + h 2 h 3 ∇·[(∇v)×(∇w)]},

where properties of the del operator were used to obtain this result. With the result

div (grad v×grad w) = 0 ,so that

∇(Fuˆeu) =

1

h 1

∂F u

∂u +Fu∇(h^2 h^3 )·

ˆeu

h 2 h 3 (See eq. (8.93)

=

1

h 1

∂F u

∂u +

Fu

h 1 h 2 h 3

∂(h 2 h 3 )

∂u =

1

h 1 h 2 h 3

∂(h 2 h 3 Fu)

∂u (See eq. (8.89)).

Similarly it can be verified that the remaining terms in equation (8.94) can be

expressed as

∇(Fvˆev) =

1

h 1 h 2 h 3

∂(h 1 h 2 Fv)

∂v

and ∇(Fweˆw) =

1

h 1 h 2 h 3

∂(h 1 h 2 Fw)

∂w.

Hence, the divergence in generalized orthogonal curvilinear coordinates can be ex-

pressed as

div F
=∇·F
=

1

h 1 h 2 h 3

[

∂(h 2 h 3 Fu)

∂u +

∂(h 1 h 3 Fv)

∂v +

∂(h 1 h 2 Fw)

∂w

]

. (8 .95)

Curl in a General Orthogonal System of Coordinates

Our problem is to derive an expression for the curlof a vector F
which is repre-

sented in the generalized orthogonal coordinates (u, v, w )in the form

F
 =F
(u, v, w ) = Fuˆeu+Fvˆev+Fwˆew

one can write

curl F
 =∇× F
 =∇× (Fuˆeu) + ∇× (Fvˆev) + ∇× (Fwˆew). (8 .96)

The first term in equation (8.96) can be expanded by using properties of the del

operator and

∇× (Fuˆeu) = ∇× (Fuh 1 ∇u) (See eq. (8.89))

=∇(Fuh 1 )×∇ u+Fuh 1 ∇×∇ u.