Geoscience Reference
In-Depth Information
In these cylindrical polar coordinates (
r
,
φ
,
z
), the gradient, divergence, curl and Laplacian
operators are
∂
T
∂
r
,
1
r
∂
T
∂φ
,
∂
T
(A1.22)
∇
T
=
∂
z
1
r
∂
∂
1
r
∂
V
φ
∂φ
+
∂
V
z
∂
(A1.23)
∇·
V
=
r
(
rV
r
)
+
z
1
r
∂
V
z
∂φ
−
∂
V
φ
∂
,
∂
V
r
∂
z
−
∂
V
z
∂
1
r
∂
∂
1
r
∂
V
r
∂φ
∇
∧
V
=
,
r
(
rV
φ
)
−
(A1.24)
z
r
r
∂
T
∂
1
r
∂
∂
r
2
∂
1
2
T
∂φ
+
∂
2
T
∂
2
T
∇
=
+
(A1.25)
r
r
2
z
2
2
V
z
V
r
r
2
r
2
∂
2
V
φ
∂φ
,
∇
r
2
∂
2
V
r
∂φ
V
φ
r
2
,
∇
2
V
2
V
r
2
V
φ
+
(A1.26)
∇
=
∇
−
−
−
Spherical polar coordinates (
r,
θ
,
φ
)
In spherical polar coordinates (Fig. A1.4), a point P is located by specifying
r
, the radius of
the sphere on which it lies,
θ
, the colatitude, and
φ
, the longitude or azimuth, where
r
≥
0,
0
≤
φ
≤
2
π
,0
≤
θ
≤
π
.From Fig. A1.4 it can be seen that
x
=
r
sin
θ
cos
φ
y
=
r
sin
θ
sin
φ
(A1.27)
θ
In spherical polar coordinates (
r
,
θ
,
φ
) the gradient, divergence, curl and Laplacian operators
are
z
=
r
cos
Figure A1.4.
∂
T
1
r
∂
T
∂θ
,
1
r
sin
∂
T
∂φ
∇
T
=
r
,
(A1.28)
∂
θ
1
r
2
∂
∂
r
(
r
2
V
r
)
+
1
r
sin
θ
∂
∂θ
1
r
sin
θ
∂
V
φ
∂φ
∇·
V
=
(sin
θ
V
θ
)
+
(A1.29)
1
r
sin
∂
∂θ
1
r
sin
∂
V
θ
∂φ
1
r
sin
∂
V
r
∂φ
∇
∧
V
=
(sin
θ
V
φ
)
−
,
θ
θ
θ
1
r
∂
∂
1
r
∂
∂
1
r
∂
V
r
∂θ
−
r
(
rV
φ
),
r
(
rV
θ
)
−
(A1.30)
r
2
sin
1
r
2
∂
∂
∂
T
1
∂
∂θ
∂
T
∂θ
1
r
2
sin
2
∂
2
T
∂φ
2
T
∇
=
+
θ
+
(A1.31)
r
∂
r
r
2
sin
θ
2
θ
2
r
2
V
r
2
∂
∂θ
2
r
2
sin
∂
V
φ
∂φ
2
V
2
V
r
∇
=
∇
−
−
(sin
θ
V
θ
)
−
,
r
2
sin
θ
θ
r
2
∂
V
r
2 cos
θ
r
2
sin
2
∂
V
φ
∂φ
,
2
V
θ
r
2
sin
2
2
V
θ
+
∇
−
−
(A1.32)
∂θ
θ
θ
2
r
2
sin
∂
V
r
∂φ
θ
r
2
sin
2
2 cos
∂
V
θ
∂φ
V
φ
r
2
sin
2
2
V
θ
+
∇
+
−
θ
θ
θ
