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

In summary, a contravariant tensor must be contracted with a covariant tensor, and vice versa, because it is a part of the definition. If one wishes to understand the more geometric picture involving vector spaces and their duals, it is suggested to refer to specific resources. Additionally, two contravariant tensors can be combined using a tensor product to create a higher rank contravariant tensor, and similarly, tensors of mixed types can also be obtained through tensor product.
  • #1
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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  • #2
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.
 
  • #3
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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Related to Must a contravariant contract with a covariant, & vice versa?

1. What is a contravariant contract?

A contravariant contract is a type of programming contract that defines the relationship between the input and output of a function. In a contravariant contract, the input type of a function can be more general than the output type.

2. What is a covariant contract?

A covariant contract is also a type of programming contract that defines the relationship between the input and output of a function. In a covariant contract, the output type of a function can be more general than the input type.

3. Can a contravariant contract be used with a covariant contract?

Yes, a contravariant contract can be used with a covariant contract. This means that the input type of one function can be more general than the output type of another function.

4. Can a covariant contract be used with a contravariant contract?

Yes, a covariant contract can also be used with a contravariant contract. This means that the output type of one function can be more general than the input type of another function.

5. Why is understanding contravariant and covariant contracts important?

Understanding contravariant and covariant contracts is important because it allows programmers to create more flexible and reusable code. By using these types of contracts, functions can accept a wider range of input and return a wider range of output, making the code more adaptable to different scenarios.

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