Begin2.DVI

Since

∂φ

∂x

eˆ 1 +∂φ

∂y

ˆe 2 +∂φ

∂z

eˆ 3 =

(

∂φ

∂u

∂u

∂x

+∂φ

∂v

∂v

∂x

+∂φ

∂w

∂w

∂x

)

ˆe 1

+

(

∂φ

∂u

∂u

∂y +

∂φ

∂v

∂v

∂y +

∂φ

∂w

∂w

∂y

)

ˆe 2

+

(

∂φ

∂u

∂u

∂z

+∂φ

∂v

∂v

∂z

+∂φ

∂w

∂w

∂z

)

ˆe 3 ,

equation (8.89) can be expressed in the form

∇φ= grad φ=∇u∂φ

∂u

+∇v∂φ

∂v

+∇w∂φ

∂w

. (8 .92)

Equation (8.92) suggests how the operator ∇can be expressed in a general curvilinear

coordinate system. In a general curvilinear coordinate system (u, v, w )one finds the

operator ∇has the form

∇=∇u∂u∂ +∇v∂v∂ +∇w∂w∂. (8 .91)

Divergence in a General Orthogonal System of Coordinates

To find the divergence in an orthogonal curvilinear system, the following rela-

tions are employed:

∇u=

1

h 1

ˆeu, ∇v=^1

h 2

eˆv, ∇w=^1

h 3

ˆew (8 .92)

which are special cases of the result in equation (8.89). Equations (8.92) imply

ˆeu=ˆev׈ew=h 2 h 3 (∇v)×(∇w)

ˆev=ˆew׈eu=h 1 h 3 (∇w)×(∇u)

ˆew=ˆeu×eˆv=h 1 h 2 (∇u)×(∇v)

(8 .93)

Example 8-19. Derive the divergence of a vector which is represented in the

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

F
=F
(u, v, w ) = Fuˆeu+Fvˆev+Fwˆew.

Solution: By using the properties of the del operator one finds

∇·F
 =∇(Fuˆeu) + ∇(Fvˆev) + ∇(Fwˆew). (8 .94)