Geoscience Reference
In-Depth Information
U
where
θ
is the angle between the two vectors and
U
and
V
are the magnitudes of the vectors
(Fig. A1.1). The scalar product is also known as the dot product. If
U
and
V
are parallel, then
θ
=
0, cos
θ
=
1, and so
U
·
V
=
1. However, if
U
and
V
are perpendicular, then
θ
=
90
◦
and
U
·
V
=
0. Thus, the scalar product of two perpendicular vectors is zero. In Cartesian
coordinates (
x
,
y
,
z
), the scalar product is
V
θ
Figure A1.1.
(A1.3)
U
·
V
=
U
x
V
x
+
U
y
V
y
+
U
z
V
z
As an example of a scalar product, consider a force
F
acting on a mass
m
and moving that
mass a distance
d
. The work done is then
F
W
·
d
:work is a scalar.
The vector product of two vectors
U
and
V
is written
U
∧
V
or
U
×
V
and defined as
U
∧
V
=
W
(A1.4)
V
θ
where
W
is a vector perpendicular both to
U
and to
V
(Fig. A1.2), with magnitude
U
(A1.5)
W
=
UV
sin
θ
Figure A1.2.
The vector product is also known as the
cross product
. The vector product of two parallel
vectors is zero since sin
θ
=
0when
θ
=
0. In Cartesian coordinates (
x
,
y
,
z
), the vector
product is expressed as
U
∧
V
=
(
U
y
V
z
−
U
z
V
y
,
U
z
V
x
−
U
x
V
z
,
U
x
V
y
−
U
y
V
x
)
(A1.6)
As an example of a vector product, consider a rigid body rotating about an axis with angular
velocity
(the Earth spinning about its north-south axis if you like). The velocity
v
of any
particle at a radial position
r
is then given by
(A1.7)
V
= ∧
r
Compare this with Eq. (2.3); rotation of the plates also involves the vector product.
Gradient
The gradient of a scalar
T
is a vector that describes the rate of change of
T
. The component of
grad
T
in any direction is the rate of change of
T
in that direction. Thus, the
x
component is
∂
T
/∂
x
, the
y
component is
∂
T
/∂
y
, and the
z
component is
∂
T
/∂
z
.grad
T
is an abbreviation for
'the gradient of
T
':
(A1.8)
grad
T
≡∇
T
defines the vector operator
∇
. The notations grad
T
and
∇
T
are equivalent and are used
interchangeably. In Cartesian coordinates (
x
,
y
,
z
),
∇
T
is given by
∂
T
∂
x
,
∂
T
∂
y
,
∂
T
(A1.9)
∇
T
=
∂
z
grad
T
is normal (perpendicular) to surfaces of constant
T
.Toshow this, consider the
temperature
T
at point (
x
,
y
,
z
). A small distance
r
=
(
x,
y,
z) away, the temperature is
T
+
T
,where
∂
T
+
∂
T
∂
+
∂
T
T
=
x
x
y
y
z
z
∂
∂
(A1.10)
=
(
∇
T
)
·
r
