I think this may be a simple yes or no question. I am currently reading a book Vector and Tensor Analysis by Borisenko. In it he introduces a reciprocal basis [itex]\vec{e_{i}}[/itex] (where i=1,2,3) for a basis [itex]\vec{e^{i}}[/itex] (where i is an index, not an exponent) that may or may not be orthogonal, normalized, and coplanar. With this is in mind he shows that a vector [itex]\vec{A}[/itex] using the properties:(adsbygoogle = window.adsbygoogle || []).push({});

[itex]\vec{A} \cdot \vec{e_{i}} = A_{i}[/itex]

[itex]\vec{A} \cdot \vec{e^{i}} = A^{i}[/itex]

may be written in the following ways...

[itex]\vec{A} = A^{1} \vec{e_{1}}+A^{2}\vec{e_{2}}+A^{3}\vec{e_{3}}[/itex]

[itex]\vec{A}=A_{1}\vec{e^{1}}+A_{2}\vec{e^{2}}+A_{3}\vec{e^{3}}[/itex]

I understand this so far...however he then seems to call the components [itex]A^{i}[/itex] the contravariant components and [itex]A_{i}[/itex] the covariant components without any particular reason as to why which is which. What significance does one have over the other in order to point out which is which?

Are the contravariant components just defined as [itex]\vec{A}[/itex] projected onto the reciprocal basis? In which case the decision as to which is the contravariant components and which is the covariant components depend on which basis was the "original?" in this case [itex]\vec{e^{i}}[/itex]?

From my searching through the threads on this site, I understand contravariant and covariant vectors go into so pretty different territory regarding manifolds and how vectors deal with certain transformations, are these applications using the same meaning that the book is using?

Thanks for the help.

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# Covariant and Contravariant Components of a Vector

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