Relative scalar: Difference between revisions

Revision as of 22:49, 3 October 2012

Revision as of 19:48, 21 February 2013

Line 15:

: <math> J = \begin{vmatrix} \displaystyle \frac{\partial(x_1,\ldots,x_n)}{\partial(\bar{x}^1,\ldots,\bar{x}^n)} \end{vmatrix} , </math>

that is, the determinant of the [[Jacobian]] of the transformation. <ref name=lovelock>{{cite book |last1=Lovelock |first1=David |last2=Rund |first2=Hanno |authorlink2=Hanno Rund |title=Tensors, Differential Forms, and Variational Principles |date=1 April 1989 | publisher=Dover | isbn=0-486-65840-6 | url=http://store.doverpublications.com/0486658406.html | accessdate=19 April 2011 | format=Paperback | chapter=4 | page=103}}</ref> A '''scalar density''' refers to the <math>w=1</math> case.

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

==Ordinary scalar==

Line 67 ⟶ 69:

:<math> \int_0^2 \! \int_{0}^{\pi/2} \! \int_0^2 \! \bar{f}(r,t,h) r \, dh \, dt \, dr = 24 + 40 \pi /3</math>,<ref>[http://www.wolframalpha.com/input/?i=int_0^2+int_0^%28pi%2F2%29+int_0^2+%282+r+cos%28t%29+%2B+r+cos%28t%29+%2B+5%29+r^2+dh+dt+dr]</ref>

which is not equal to the previous case.

==Scalar density==

A '''scalar density''' refers to the <math>w=1</math> 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 tensor]]s.

==See also==