Tensor calculus> definition of contravariants

[Abolaban]

Hello Big minds,

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_transformationplease note that I newly "ride on my horse" through tensor analysis!best regardsAbolaban

 

[Orodruin]

General coordinate transformations are not necessarily rotations. When you deal with Cartesian tensors, you will only come across rotations and so your transformations will contain sines and cosines. However, the general case is more ... well ... general.

When you refer to pages in Arfken, please also state the edition - there are now seven of them ...

 

[Abolaban]

thank you "Orodruin" for your answer...

I have the sixth ed...

Is that defenition of Arfken Contravariant vector -and hence tensor- consistent with the following definition:
http://www.encyclopediaofmath.org/index.php/Contravariant_tensor

best regardsAbolaban

 

[Abolaban]

that was from Arfken's book 6th ed

 

[Orodruin]

Yes, they are compatible. The difference is that Arfken's definition is a more pragmatic one based on the important physics properties of tensors while the other would satisfy a mathematician to a higher degree.