Geoscience Reference
In-Depth Information
Appendix 1
Scalars, vectors and differential
operators
Scalars and vectors
A scalar is a quantity that just has a magnitude. For example, the temperature outside today
could be
+
10
◦
C.
Avector is a quantity that has a magnitude and a direction. For example, the wind velocity
in your city today could be 20 km hr
−
1
and due east. The speed (magnitude of the velocity)
would be measured by an anemometer and the direction by a wind vane.
Avector is indicated in print by a boldface character such as
x
. Its magnitude is indicated
by the same character in italic type,
x
,asisascalar. A vector or a scalar can be either a
constant or a function of some variable, which can itself be either a scalar or a vector. When
the scalar (or vector) is a function of a variable, it is called a scalar (or vector) field. For
example, the temperature at midday across the province of British Columbia, Canada, is a
scalar field. That is, the temperature at each place depends on its position in the province and
thus is written
T
(
x
,
y
,
z
), where
x
,
y
and
z
are geographic and height coordinates within British
Columbia. The wind velocity
V
across British Columbia at midday depends on geographic
position and so is a vector field, written
V
(
x
z
). Note that, if the coordinate system is
changed, the scalar is unaffected, but the components of the vector must be recalculated.
,
y
,
Products of scalars and vectors
Many physical relationships are best expressed by using the products of scalars and vectors.
The product of two scalars is another scalar, and everyone is well accustomed to the process
called multiplication, learned laboriously in elementary school. The product of a scalar
s
and
avector
V
=
(
V
x
,
V
y
,
V
z
)isanother vector,
s
V
.InCartesian coordinates (
x
,
y
,
z
), the product
is simply
s
V
=
s
(
V
x
,
V
x
,
V
z
)
=
(
sV
x
,
sV
y
,
sV
z
)
(A1.1)
Thus, the scalar multiplies each component of the vector.
When two vectors are involved, multiplication becomes more complicated. There are two
products of vectors: one, called the
scalar product
,isascalar; the other, called the
vector
product
,isavector.
The scalar product of two vectors
U
and
V
is written
U
·
V
and defined as
U
·
V
=
UV
cos
θ
(A1.2)
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