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Jason Quinn (talk | contribs) →Ordinary scalar: improved Veblen ref |
Jason Quinn (talk | contribs) →Ordinary scalar: +url to veblen ref |
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==Ordinary scalar==
An '''ordinary scalar''' or '''absolute scalar'''<ref>{{cite book |last=Veblen |first=Oswald |authorlink=Oswald Veblen |title=Invariants of Quadratic Differential Forms |url=http://www.cambridge.org/us/knowledge/isbn/item1156775/?site_locale=en_US |accessdate=3 October 2012 |year=2004 |publisher=Cambridge University Press |isbn=0-521-60484-2 |page=21}}</ref> refers to the <math>w=0</math> case.
If <math>x^i</math> and <math>\bar{x}^j</math> refer to the same point <math>P</math> on the manifold, then we desire <math>\bar{f}(\bar{x}^j) = f(x^i)</math>. This equation can be interpreted two ways when <math>\bar{x}^j</math> are viewed as the "new coordinates" and <math>x^i</math> are viewed as the "original coordinates". The first is as <math>\bar{f}(\bar{x}^j) = f(x^i(\bar{x}^j))</math>, which "converts the function to the new coordinates". The second is as <math>f(x^i)=\bar{f}(\bar{x}^j(x^i))</math>, which "converts back to the original coordinates. Of course, "new" or "original" is a relative concept.
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