Geology Reference
In-Depth Information
Appendix 1: Vector Calculus
If
r
and
r
C
d
r
are two neighbor points on a
contour line, then it results:
A1.1 Scalar Fields
Many physical quantities can be represented by
a single real number. They are called
scalars
.
Examples of scalar quantities are the temperature,
T
, the pressure,
P
, and the density, ¡. A scalar
field ¥
D
¥(
r
) is a continuous function of the
spatial Cartesian coordinates (
x
,
y
,
z
), which can be
represented by a position vector
r
D
x
i
C
y
j
C
z
k
.
The spatial variability of a scalar field can be
expressed through its
gradient
:
@¥
@x
dx
C
@¥
@y
dy
d¥
D
¥.r
C
dr/
¥.r/
D
(A1.2)
Dr
¥
dr
D
0
Therefore,
r
¥ is always orthogonal to the con-
tour lines of a scalar field in the plane. This im-
plies that the direction of
r
¥ coincides with the
direction of maximum
increase
of ¥ (Fig.
A1.3
).
Similarly, in the general case of three-
dimensional fields, the set of points (
x
,
y
,
z
)such
that ¥(
x
,
y
,
z
)
D
¥
0
@¥
@x
;
@¥
@y
;
@¥
@¥
@x
i
C
@¥
@y
j
C
@¥
@
z
k
(A1.1)
r
¥
@
z
3
.In
this instance,
r
¥ is always orthogonal to the
isosurfaces of ¥. Given a direction unit vector
n
,
the quantity:
is
an isosurface in
<
The gradient of a scalar field is not a scalar
quantity, because it is formed by the ordered set
of three spatial derivatives. It is an example of
vector field
, which defines a vector quantity for
each position (
x
,
y
,
z
). In this instance, the vec-
tor components are the three derivatives @¥/@
x
,
@¥/@
y
,and@¥/@
z
. Many scalar fields considered
in geophysics are functions of only two spatial
coordinates, for example latitude and longitude,
thereby they can be represented as surfaces on the
plane or on the sphere (Fig.
A1.1
).
Elevation, gravity and magnetic anomalies,
and surface heat flux are examples of scalar
fields in the plane. In this instance, the equation
¥
D
¥
0
, ¥
0
being a constant field value, defines
a
contour line
for the scalar field in the (
x
,
y
)
plane. This is the set of points (
x
,
y
) such that
¥(
x
,
y
)
D
¥
0
. A set of contour lines for equally
spaced values ¥
0
, ¥
1
, ::: is a convenient way to
illustrate the field properties, alternative to the
surface representation (Fig.
A1.2
).
@¥
@n
D
@¥
@x
n
x
C
@¥
@y
n
y
C
@¥
@
z
n
z
(A1.3)
r
¥
n
is called
directional derivative
of ¥ in the direc-
tion of
n
. It represents the variation of ¥ as we
move from the actual position
r
to a neighbor
position
r
C
d
r
in the direction
n
.
A1.2 Vector Fields
A vector field
A
D
A
(
r
) associates a vector
A
to
each position
r
(
x
,
y
,
z
) in the space. Classic ex-
amples are the force fields, in particular electric,
gravity, and magnetic fields, but also velocity
and acceleration fields. The spatial variability of
vector fields is described through two differential
