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
that is, the determinant of the Jacobian of the transformation.
[1] A scalar density refers to the
Relative scalars are an important special case of the more general concept of a relative tensor.
This equation can be interpreted two ways when
, which "converts back to the original coordinates.
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
The two coordinate systems are related by the following sets of equations:
A quick calculation shows that
This equality would have held for any chosen point
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.
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.)
(that is, the "room" is a quarter slice of a cylinder of radius and height 2).
over the same region is[citation needed]
The integral of temperature is not independent of the coordinate system used.
It is non-physical in that sense, hence "strange".
included a factor of the Jacobian (which is just
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.
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.
As before is integral (the total mass) in Cartesian coordinates is
also included a factor of the Jacobian like before, we get[citation needed]
which is not equal to the previous case.
It can be shown the determinant of a type (0,2) tensor is a relative scalar of weight 2.