Relative scalar

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

$\bar{x}^j = \bar{x}^j(x^i)$

on an n-dimensional manifold obeys the following equation

$\bar{f}(\bar{x}^j) = J^w f(x^i)$

where

$J = \begin{vmatrix} \displaystyle \frac{\partial(x_1,\ldots,x_n)}{\partial(\bar{x}^1,\ldots,\bar{x}^n)} \end{vmatrix} ,$

that is, the determinant of the Jacobian of the transformation.^[1] Relative scalars are an important special case of the more general concept of a relative tensor.

1 Ordinary scalar
- 1.1 Elementary example
2 Scalar density
3 Other cases
4 Generalization
5 See also
6 References

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.

Elementary example

Suppose the temperature in a room is given in terms of the function $f(x,y,z)= 2 x %2B y %2B 5$ 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:

$r = \sqrt{x^2 %2B y^2} \,$

$t = \arctan(y/x) \,$

$h = z \,$

and

$x = r \cos(t) \,$

$t = r \sin(t) \,$

$z = h. \,$

Using $\bar{f}(\bar{x}^j) = f(x^i(\bar{x}^j))$ allows one to derive $\bar{f}(r,t,h)= 2 r \cos(t)%2B r \sin(t) %2B 5$ 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^{[says who?]} 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

$\int_0^2 \! \int_{-\sqrt{2^2-x^2}}^\sqrt{2^2-x^2} \! \int_0^2 \! f(x,y,z) \, dz \, dy \, dx = 64/3 %2B 20 \pi$ .

The value of the integral of $\bar{f}$ over the same region is

$\int_0^2 \! \int_{0}^{\pi/2} \! \int_0^2 \! \bar{f}(r,t,h) \, dh \, dt \, dr = 12 %2B 10 \pi$ .

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.

Scalar density

A scalar density refers to the $w=1$ 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.

Generalization

Relative scalars are special cases of relative tensors.

References

^ Lovelock, David; Rund (April 1, 1989). "4" (Paperback). Tensors, Differential Forms, and Variational Principles. Dover. p. 103. ISBN 0486658406. http://store.doverpublications.com/0486658406.html. Retrieved 19 April 2011.
^ Veblen, Oswald (2004). Invariants of Quadratic Differential Forms. Cambridge University Press. pp. 112. ISBN 0521604842. (page 21)