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In the book of Arfken [Math Meth for Physicists] p 134 he defined contravariant tensor...my question is about a_ij he defined them first as cosines of an angle of basis then he suddenly replaced them by differential notation...why is that?

cosines are not mention in this article as well:

http://en.wikipedia.org/wiki/Covariant_transformation

please note that I newly "ride on my horse" through tensor analysis!

best regards

Abolaban