Relative scalar
Encyclopedia
In mathematics, a relative scalar (of weight w) is a scalar-valued function whose transform under a coordinate transform,
on an n-dimensional manifold obeys the following equation
where
that is, the determinant of the Jacobian of the transformation. Relative scalars are an important special case of the more general concept of a relative tensor.
case.
If
and 
refer to the same point 
on the manifold, then we desire 
. This equation can be interpreted two ways when 
are viewed as the "new coordinates" and 
are viewed as the "original coordinates". The first is as 
, which "converts the function to the new coordinates". The second is as 
, 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.
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:
and
Using
allows one to derive 
as the transformed function.
Consider the point
whose Cartesian coordinates are 
and whose corresponding value in the cylindrical system is 
. A quick calculation shows that 
and 
also. This equality would have held for any chosen point P. Thus, 
is the "temperature function in the Cartesian coordinate system" and 
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
and wished to have derived the Cartesian temperature function 
. 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
. (Yes, integrating temperature is strange but that's partly what's to be shown.) Suppose the region 
is given in cylindrical coordinates as 
from 
, 
from 
and 
from 
(that is, the "room" is a quarter slice of a cylinder of radius and height 2).
The integral of
over the region 
is
.
The value of the integral of
over the same region is
.
They are not equal. The integral of an ordinary scalar depends on the coordinate system used. This coordinate dependence tends to remove any physical meaning from the integral of an ordinary scalar.
on an n-dimensional manifold obeys the following equation
where
that is, the determinant of the Jacobian of the transformation. Relative scalars are an important special case of the more general concept of a relative tensor.
Ordinary scalar
An ordinary scalar or absolute scalar refers to the case.If
and refer to the same point on the manifold, then we desire . This equation can be interpreted two ways when are viewed as the "new coordinates" and are viewed as the "original coordinates". The first is as , which "converts the function to the new coordinates". The second is as , 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.
Elementary example
Suppose the temperature in a room is given in terms of the function 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:and
Using
allows one to derive as the transformed function.Consider the point
whose Cartesian coordinates are and whose corresponding value in the cylindrical system is . A quick calculation shows that and also. This equality would have held for any chosen point P. Thus, is the "temperature function in the Cartesian coordinate system" and 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
and wished to have derived the Cartesian temperature function . 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
. (Yes, integrating temperature is strange but that's partly what's to be shown.) Suppose the region is given in cylindrical coordinates as from , from and from (that is, the "room" is a quarter slice of a cylinder of radius and height 2).The integral of
over the region is.The value of the integral of
over the same region is.They are not equal. The integral of an ordinary scalar depends on the coordinate system used. This coordinate dependence tends to remove any physical meaning from the integral of an ordinary scalar.