Must a contravariant contract with a covariant, & vice versa?

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SUMMARY

A contravariant tensor must be contracted with a covariant tensor due to the inherent definitions and transformation properties of these tensors. The discussion emphasizes the importance of understanding contracted tensor products to grasp why this relationship exists. For a deeper comprehension, readers are encouraged to explore geometric perspectives involving vector spaces and their duals, as outlined in the suggested resource.

PREREQUISITES
  • Understanding of contravariant and covariant tensors
  • Familiarity with tensor products and their properties
  • Basic knowledge of vector spaces and dual spaces
  • Ability to interpret tensor transformation properties
NEXT STEPS
  • Study the concept of contracted tensor products in detail
  • Explore the geometric interpretation of vector spaces and duals
  • Learn about tensor products of mixed types and their applications
  • Review transformation properties of tensors in advanced mathematics
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Mathematicians, physicists, and students studying advanced topics in tensor analysis and differential geometry.

Master J
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Why is it that a contravariant tensor must be contracted with a covariant tensor, and vice versa? Why is this so?
 
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You can't justify a definition, just read it carefully. So reread what a contracted tensor product is and you'll understand what exactly happens.
 
As said it is a part of the definition. If you time and want to learn the more geometric picture involving vector spaces and their duals then I suggest for instance as a start

http://www.strw.leidenuniv.nl/~yuri/GR/handout1.pdf

This type of perspective will certainly make more sense and you will understand why they are defined by their transformation properties.

ps: what you can do with two contravariant tensors is for instance to take their tensor product to obtain another contravariant tensor of higher rank (that is with more indices). Similiarly you may take tensor product of two tensors of any type and possibly obtain tensors of mixed type too.
 
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