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.[1] A scalar density refers to the case.

Relative scalars are an important special case of the more general concept of a relative tensor.

Ordinary scalar

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An ordinary scalar or absolute scalar[2] 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.

Weight 0 example

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Suppose the temperature in a room is given in terms of the function   in Cartesian coordinates   and the function in cylindrical coordinates   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  . 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[citation needed]   The value of the integral of   over the same region is[citation needed]   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   included a factor of the Jacobian (which is just  ), we get[citation needed]   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

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If we had said   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  . This time   but  . As before is integral (the total mass) in Cartesian coordinates is   The value of the integral of   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   also included a factor of the Jacobian like before, we get[citation needed]   which is not equal to the previous case.

Other cases

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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.

See also

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References

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  1. ^ Lovelock, David; Rund, Hanno (1 April 1989). "4". Tensors, Differential Forms, and Variational Principles (Paperback). Dover. p. 103. ISBN 0-486-65840-6. Retrieved 19 April 2011.
  2. ^ Veblen, Oswald (2004). Invariants of Quadratic Differential Forms. Cambridge University Press. p. 21. ISBN 0-521-60484-2. Retrieved 3 October 2012.