In mathematics, a relative scalar (of weight w) is a scalar-valued function whose transform under a coordinate transform,
x
¯
j
=
x
¯
j
(
x
i
)
on an n-dimensional manifold obeys the following equation
f
¯
(
x
¯
j
)
=
J
w
f
(
x
i
)
where
J
=
|
∂
(
x
1
,
…
,
x
n
)
∂
(
x
¯
1
,
…
,
x
¯
n
)
|
,
that is, the determinant of the Jacobian of the transformation. A scalar density refers to the
w
=
1
case.
Relative scalars are an important special case of the more general concept of a relative tensor.
An ordinary scalar or absolute scalar refers to the
w
=
0
case.
If
x
i
and
x
¯
j
refer to the same point
P
on the manifold, then we desire
f
¯
(
x
¯
j
)
=
f
(
x
i
)
. This equation can be interpreted two ways when
x
¯
j
are viewed as the "new coordinates" and
x
i
are viewed as the "original coordinates". The first is as
f
¯
(
x
¯
j
)
=
f
(
x
i
(
x
¯
j
)
)
, which "converts the function to the new coordinates". The second is as
f
(
x
i
)
=
f
¯
(
x
¯
j
(
x
i
)
)
, which "converts back to the original coordinates. Of course, "new" or "original" is a relative concept.
There are many physical quantities that are represented by ordinary scalars, such as temperature and pressure.
Suppose the temperature in a room is given in terms of the function
f
(
x
,
y
,
z
)
=
2
x
+
y
+
5
in Cartesian coordinates
(
x
,
y
,
z
)
and the function in cylindrical coordinates
(
r
,
t
,
h
)
is desired. The two coordinate systems are related by the following sets of equations:
r
=
x
2
+
y
2
t
=
arctan
(
y
/
x
)
h
=
z
and
x
=
r
cos
(
t
)
y
=
r
sin
(
t
)
z
=
h
.
Using
f
¯
(
x
¯
j
)
=
f
(
x
i
(
x
¯
j
)
)
allows one to derive
f
¯
(
r
,
t
,
h
)
=
2
r
cos
(
t
)
+
r
sin
(
t
)
+
5
as the transformed function.
Consider the point
P
whose Cartesian coordinates are
(
x
,
y
,
z
)
=
(
2
,
3
,
4
)
and whose corresponding value in the cylindrical system is
(
r
,
t
,
h
)
=
(
13
,
arctan
(
3
/
2
)
,
4
)
. A quick calculation shows that
f
(
2
,
3
,
4
)
=
12
and
f
¯
(
13
,
arctan
(
3
/
2
)
,
4
)
=
12
also. This equality would have held for any chosen point
P
. Thus,
f
(
x
,
y
,
z
)
is the "temperature function in the Cartesian coordinate system" and
f
¯
(
r
,
t
,
h
)
is the "temperature function in the cylindrical coordinate system".
One way to view these functions is as representations of the "parent" function that takes a point of the manifold as an argument and gives the temperature.
The problem could have been reversed. One could have been given
f
¯
and wished to have derived the Cartesian temperature function
f
. This just flips the notion of "new" vs the "original" coordinate system.
Suppose that one wishes to integrate these functions over "the room", which will be denoted by
D
. (Yes, integrating temperature is strange but that's partly what's to be shown.) Suppose the region
D
is given in cylindrical coordinates as
r
from
[
0
,
2
]
,
t
from
[
0
,
π
/
2
]
and
h
from
[
0
,
2
]
(that is, the "room" is a quarter slice of a cylinder of radius and height 2). The integral of
f
over the region
D
is
∫
0
2
∫
0
2
2
−
x
2
∫
0
2
f
(
x
,
y
,
z
)
d
z
d
y
d
x
=
16
+
10
π
.
The value of the integral of
f
¯
over the same region is
∫
0
2
∫
0
π
/
2
∫
0
2
f
¯
(
r
,
t
,
h
)
d
h
d
t
d
r
=
12
+
10
π
.
They are not equal. The integral of temperature is not independent of the coordinate system used. It is non-physical in that sense, hence "strange". Note that if the integral of
f
¯
included a factor of the Jacobian (which is just
r
), we get
∫
0
2
∫
0
π
/
2
∫
0
2
f
¯
(
r
,
t
,
h
)
r
d
h
d
t
d
r
=
16
+
10
π
,
which is equal to the original integral but it is not however the integral of temperature because temperature is a relative scalar of weight 0, not a relative scalar of weight 1.
If we had said
f
(
x
,
y
,
z
)
=
2
x
+
y
+
5
was representing mass density, however, then its transformed value should include the Jacobian factor that takes into account the geometric distortion of the coordinate system. The transformed function is now
f
¯
(
r
,
t
,
h
)
=
(
2
r
cos
(
t
)
+
r
sin
(
t
)
+
5
)
r
. This time
f
(
2
,
3
,
4
)
=
12
but
f
¯
(
13
,
arctan
(
3
/
2
)
,
4
)
=
12
29
. As before is integral (the total mass) in Cartesian coordinates is
∫
0
2
∫
0
2
2
−
x
2
∫
0
2
f
(
x
,
y
,
z
)
d
z
d
y
d
x
=
16
+
10
π
.
The value of the integral of
f
¯
over the same region is
∫
0
2
∫
0
π
/
2
∫
0
2
f
¯
(
r
,
t
,
h
)
d
h
d
t
d
r
=
16
+
10
π
.
They are equal. The integral of mass density gives total mass which is a coordinate-independent concept. Note that if the integral of
f
¯
also included a factor of the Jacobian like before, we get
∫
0
2
∫
0
π
/
2
∫
0
2
f
¯
(
r
,
t
,
h
)
r
d
h
d
t
d
r
=
24
+
40
π
/
3
,
which is not equal to the previous case.
Weights other than 0 and 1 do not arise as often. It can be shown the determinant of a type (0,2) tensor is a relative scalar of weight 2.
