Relative scalar explained

In mathematics, a relative scalar (of weight w) is a scalar-valued function whose transform under a coordinate transform,

$$\backslash bar^j\; =\; \backslash bar^j(x^i)$$

on an n-dimensional manifold obeys the following equation

$$\backslash bar(\backslash bar^j)\; =\; J^w\; f(x^i)$$

where

$$J\; =\; \backslash left|\; \backslash dfrac\; \backslash right|,$$

that is, the determinant of the Jacobian of the transformation.^[1] 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.

Ordinary scalar

An ordinary scalar or absolute scalar^[2] refers to the

w =0 case.

If

x ^{i} and \bar{x} ^{j} refer to the same point P on the manifold, then we desire \bar{f}(\bar{x} ^{j)} = f(x ^{i)} . This equation can be interpreted two ways when \bar{x} ^{j} are viewed as the "new coordinates" and x ^{i} are viewed as the "original coordinates". The first is as \bar{f}(\bar{x} ^{j)} = f(x ^{i(\bar{x}} ^{j))} , which "converts the function to the new coordinates". The second is as f(x ^{i)=\bar{f}(\bar{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.

Weight 0 example

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:and

Using

\bar{f}(\bar{x} ^{j)} = f(x ^{i(\bar{x}} ^{j))} allows one to derive \bar{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) = (\sqrt{ 13 },\arctan{( 3 / 2 )}, 4 ) . A quick calculation shows that f( 2 , 3 , 4 ) =12 and \bar{f}(\sqrt{ 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 \bar{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

\bar{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 ,\pi/ 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$$\backslash int\_0^2\; \backslash !\; \backslash int\_^\backslash sqrt\; \backslash !\; \backslash int\_0^2\; \backslash !\; f(x,y,z)\; \backslash ,\; dz\; \backslash ,\; dy\; \backslash ,\; dx\; =\; 16\; +\; 10\; \backslash pi.$$The value of the integral of \bar{f} over the same region is$$\backslash int\_0^2\; \backslash !\; \backslash int\_^\; \backslash !\; \backslash int\_0^2\; \backslash !\; \backslash bar(r,t,h)\; \backslash ,\; dh\; \backslash ,\; dt\; \backslash ,\; dr\; =\; 12\; +\; 10\; \backslash pi.$$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 \bar{f} included a factor of the Jacobian (which is just r ), we get$$\backslash int\_0^2\; \backslash !\; \backslash int\_^\; \backslash !\; \backslash int\_0^2\; \backslash !\; \backslash bar(r,t,h)\; r\; \backslash ,\; dh\; \backslash ,\; dt\; \backslash ,\; dr\; =\; 16\; +\; 10\; \backslash pi,$$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

f(x,y,z) = 2 x + y + 5 was representing mass density, however, then its transformed valueshould include the Jacobian factor that takes into account the geometric distortion of the coordinatesystem. The transformed function is now \bar{f}(r,t,h) = ( 2 r \cos(t) + r \sin(t) + 5 ) r . This timef( 2 , 3 , 4 ) =12 but \bar{f}(\sqrt{ 13 },\arctan{( 3 / 2 )}, 4 ) =12 \sqrt{ 29 } . As beforeis integral (the total mass) in Cartesian coordinates is$$\backslash int\_0^2\; \backslash !\; \backslash int\_^\backslash sqrt\; \backslash !\; \backslash int\_0^2\; \backslash !\; f(x,y,z)\; \backslash ,\; dz\; \backslash ,\; dy\; \backslash ,\; dx\; =\; 16\; +\; 10\; \backslash pi.$$The value of the integral of \bar{f} over the same region is$$\backslash int\_0^2\; \backslash !\; \backslash int\_^\; \backslash !\; \backslash int\_0^2\; \backslash !\; \backslash bar(r,t,h)\; \backslash ,\; dh\; \backslash ,\; dt\; \backslash ,\; dr\; =\; 16\; +\; 10\; \backslash pi.$$They are equal. The integral of mass density gives total mass which is a coordinate-independent concept.Note that if the integral of \bar{f} also included a factor of the Jacobian like before, we get$$\backslash int\_0^2\; \backslash !\; \backslash int\_^\; \backslash !\; \backslash int\_0^2\; \backslash !\; \backslash bar(r,t,h)\; r\; \backslash ,\; dh\; \backslash ,\; dt\; \backslash ,\; dr\; =\; 24\; +\; 40\; \backslash pi\; /\; 3,$$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.

See also

• Jacobian matrix and determinant

Notes and References

1. Book: Lovelock . David . Rund . Hanno . Hanno Rund . Tensors, Differential Forms, and Variational Principles . 1 April 1989 . Dover . 0-486-65840-6 . 19 April 2011 . Paperback . 4 . 103.
2. Book: Veblen, Oswald . Oswald Veblen

   . Oswald Veblen . Invariants of Quadratic Differential Forms . 3 October 2012 . 2004 . . 0-521-60484-2 . 21.