Begin2.DVI

Since curl grad u= 0,the above simplifies to

∇× (Fuˆeu) = ∇(Fuh 1 )×

ˆeu

h 1

=

[

1

h 1

∂(Fuh 1 )

∂u

ˆeu+^1

h 2

∂(Fuh 1 )

∂v

ˆev+^1

h 3

∂(Fuh 1 )

∂w

eˆw

]

× ˆeu

h 1

=

1

h 1 h 3

∂(Fuh 1 )

∂w

ˆev−^1

h 1 h 2

∂(Fuh 1 )

∂v

ˆew

In a similar manner it may be verified that the remaining terms in equation (8.96)

can be expressed as

∇× (Fvˆev) =^1

h 1 h 2

∂(Fvh 2 )

∂u

ˆew−^1

h 2 h 3

∂(Fvh 2 )

∂w

ˆeu

and ∇× (Fwˆew) =^1

h 2 h 3

∂(Fwh 3 )

∂v

ˆeu−^1

h 1 h 3

∂(Fwh 3 )

∂u

ˆev.

Hence, the curlof a vector in generalized curvilinear coordinates can be represented

in the form

∇× F
=^1

h 2 h 3

[

∂(Fwh 3 )

∂v

−∂(Fvh^2 )

∂w

]

ˆeu

+^1

h 1 h 3

[

∂(Fuh 1 )

∂w

−∂(Fwh^3 )

∂u

]

ˆev

+

1

h 1 h 2

[

∂(Fvh 2 )

∂u −

∂(Fuh 1 )

∂v

]

ˆew.

(8 .97)

Equation (8.97) can also be represented in the determinant form as

∇× F
=h^1

1 h 2 h 3

∣∣

∣∣

∣∣

h 1 ˆeu h 2 ˆev h 3 ˆew

∂

∂u

∂

∂v

∂

∂w

Fuh 1 Fvh 2 Fwh 3

∣∣

∣∣

∣∣. (8 .98)

The Laplacian in Generalized Orthogonal Coordinates

Using the definition ∇^2 φ=∇∇ φand the relation for the gradient given by equa-

tion (8.89) and show that

∇∇ φ=∇

[

1

h 1

∂φ

∂u eˆu+

1

h 2

∂φ

∂v ˆev+

1

h 3

∂φ

∂w ˆew

]

. (8 .99)

The result of equation (8.95) simplifies equation (8.99) to the final form given as

∇^2 φ=

1

h 1 h 2 h 3

[

∂

∂u

(

h 2 h 3

h 1

∂φ

∂u

)

=

∂

∂v

(

h 1 h 3

h 2

∂φ

∂v

)

+

∂

∂w

(

h 1 h 2

h 3

∂φ

∂w

)]

. (8 .100)

The equation ∇^2 U= 0 is known as Laplace’s equation.