Notes 2
Fall 2004.doc
Scalar Fields and Vector Fields
A voltage as a function of time may be written as
( )
t
v
.
This is an example of a function of one
variable.
In electromagnetics this, as well as other quantities of interest, are also often functions
of time
and
space.
In the Cartesian coordinate system we might write
( )
t
,
z
,
y
,
x
v
v
→
The fact that this quantity is now a function of three dimensional space makes this a
field
.
The
fact that this quantity is a scalar (there is no direction associated with it) as opposed to a vector,
makes this a
scalar field
.
Another good example of a scalar field is the temperature at different
locations in the atmosphere.
A hot air balloonist, for instance, traveling between Baton Rouge
and New Orleans might be concerned about the temperatures at different altitudes
( )
z
, at
different points,
along his route.
(
y
x
,
)
Since vector quantities are often of interest (force, velocity, momentum, current density, electric
and magnetic fields …) it is reasonable to want to consider
vector fields,
( )
t
,
z
,
y
,
x
E
E
G
G
→
G
Choosing the symbol
E
for this example is appropriate because we are thinking about the
electric field which is a vector quantity with both a magnitude and direction.
Staying with the
Cartesian coordinate system we can decompose this vector field into its components like this,
()
( ) (
)(
z
ˆ
t
,
z
,
y
,
x
y
ˆ
t
,
z
,
y
,
x
x
ˆ
t
,
z
,
y
,
x
t
,
z
,
y
,
x
z
y
x
E
E
E
E
)
+
+
=
G
The three components of
,
,
,
are themselves, scalar fields, and so we can say that to
describe the information in a vector field we need to keep track of three scalar fields.
,
, and
are
unit vectors
(of length one) and balance the vector nature of the left hand side of the
equation with the right hand side.
E
G
x
E
y
E
z
E
x
ˆ
y
ˆ
z
ˆ
Taking derivatives of fields:
The del operator, the gradient, the divergence
and the curl.
Differentiating scalar and vector fields:
()
[]
( )
t
t
,
z
,
y
,
x
t
,
z
,
y
,
x
dt
d
∂
∂
=
v
v
()
[]
( )
( )
( )
z
ˆ
t
t
,
z
,
y
,
x
y
ˆ
t
t
,
z
,
y
,
x
x
ˆ
t
t
,
z
,
y
,
x
t
,
z
,
y
,
x
dt
d
z
y
x
∂
∂
+
∂
∂
+
∂
∂
=
E
E
E
E
G
1