where Au, A v, A w denote the components of A in the new coordinate system and
are functions of these coordinates. The transformation equations from rectangular
coordinates to curvilinear coordinates is represented by the matrix equation
Au
Av
Aw
=
eˆ 1 ·ˆeu eˆ 2 ·ˆeu eˆ 3 ·ˆeu
ˆe 1 ·ˆev ˆe 2 ·ˆev ˆe 3 ·ˆew
ˆe 1 ·ˆew ˆe 2 ·ˆew ˆe 3 ·ˆew
A 1
A 2
A 3
(8 .87)
which is derived by taking projections of the vector Aonto the u,vand wdirections.
Let us find the representation of the gradient, divergence and curl in a general
orthogonal curvilinear coordinate system. Recall that the gradient, divergence, and
curlin rectangular coordinates are given by
grad φ=
∂φ
∂x
eˆ 1 +∂φ
∂y
ˆe 2 +∂φ
∂z
ˆe 3
div F =
∂F 1
∂x +
∂F 2
∂y +
∂F 3
∂z
curl F =
(
∂F 3
∂y −
∂F 2
∂z
)
eˆ 1 +
(
∂F 1
∂z −
∂F 3
∂x
)
ˆe 2 +
(
∂F 2
∂x −
∂F 1
∂y
)
ˆe 3.
Gradient in a General Orthogonal System of Coordinates
In an orthogonal curvilinear coordinate system, let the vector grad φhave the
representation
grad φ=Auˆeu+Aveˆv+Awˆew.
By using the matrix equation (8.87), the component Auin the curvilinear coordinates
is
grad φ·ˆeu=Au=∂φ
∂x
ˆe 1 ·ˆeu+∂φ
∂y
ˆe 2 ·ˆeu+∂φ
∂z
ˆe 3 ·ˆeu.
By employing equations (8.75) and (8.76), this result simplifies and
Au=
1
h 1
[
∂φ
∂x
∂x
∂u +
∂φ
∂y
∂y
∂u +
∂φ
∂z
∂z
∂u
]
=
1
h 1
∂φ
∂u. (8 .88)
In a similar manner, it can be shown that the other components have the form
grad φ·ˆev=Av=
1
h 2
∂φ
∂v and grad φ·ˆew=Aw=
1
h 3
∂φ
∂w.
Thus the gradient can be represented in the curvilinear coordinate system as
∇φ= grad φ=h^1
1
∂φ
∂u ˆeu+
1
h 2
∂φ
∂v ˆev+
1
h 3
∂φ
∂w eˆw. (8 .89)
