Relative scalar

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. A scalar density refers to the formula_4 case.
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 formula_5 case.
If formula_6 and formula_7 refer to the same point formula_8 on the manifold, then we desire formula_9. This equation can be interpreted two ways when formula_7 are viewed as the "new coordinates" and formula_6 are viewed as the "original coordinates". The first is as formula_12, which "converts the function to the new coordinates". The second is as formula_13, 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.
Weight 0 example.
Suppose the temperature in a room is given in terms of the function formula_14 in Cartesian coordinates formula_15 and the function in cylindrical coordinates formula_16 is desired. The two coordinate systems are related by the following sets of equations:
and
Using formula_12 allows one to derive formula_24 as the transformed function.
Consider the point formula_8 whose Cartesian coordinates are formula_26 and whose corresponding value in the cylindrical system is formula_27. A quick calculation shows that formula_28 and formula_29 also. This equality would have held for any chosen point formula_8. Thus, formula_31 is the "temperature function in the Cartesian coordinate system" and formula_32 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 formula_33 and wished to have derived the Cartesian temperature function formula_34. 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 formula_35. (Yes, integrating temperature is strange but that's partly what's to be shown.) Suppose the region formula_35 is given in cylindrical coordinates as formula_37 from formula_38, formula_39 from formula_40 and formula_41 from formula_38 (that is, the "room" is a quarter slice of a cylinder of radius and height 2).
The integral of formula_34 over the region formula_35 is
The value of the integral of formula_33 over the same region is
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 formula_33 included a factor of the Jacobian (which is just formula_37),
we get
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.
Weight 1 example.
If we had said formula_14 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 formula_52. This time
formula_28 but formula_54. As before
is integral (the total mass) in Cartesian coordinates is
The value of the integral of formula_33 over the same region is
They are equal. The integral of mass "density" gives total mass which is a coordinate-independent concept.
Note that if the integral of formula_33 also included a factor of the Jacobian like before, we get
which is not equal to the previous case.
Other cases.
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.