Covariant and Contravariant Rank-2 Tensors

In summary, the conversation is about the necessity of a metric tensor in order for a tensor to be transformed from its covariant form to its contravariant form. The concept is important in understanding tensors, and in Minkowski space, it applies to transforming matrices, not just vectors. There is also a discussion about the meaning of transforming a contravariant tensor to a covariant one, and the use of different types of multiplication in tensor equations.
  • #1
Truth Finder
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Dear Fellows,

Do anyone have an idea of whether there must be a system tensor in order to be able to transform from the covariant form of a certain tensor to its contravariant one?

This is a bit important to get rigid basics about tensors.


Schwartz Vandslire

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Either to it correctly as required, or to pass it as required.
 
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  • #2
If by "system" tensor you mean "metric" tensor, then yes, in order to have covariant and contravariant vectors and tensors, you must have a metric tensor such that ai= gijaj.

A more general Riemann space may have a "Riemann connection" rather than a metric tensor but my understanding is that we do no talk about covariant and contravariant vectors and tensors in such a space.
 
  • #3
Oh! Thanks!

But, HallsofIvy, let's talk, in special, in Minkoweski Space? I know that the latter is a physical concept, but, it refers to a special case which is 4 indices. What about transforming 4x4-Matrices (or Tensors, to be more precise), not only Vectors?

But I have a question please. Does the previous relation also apply to transforming position vectors (They are the basics of the coordinate system)?

What is the meaning of transforming a contravariant tensor to a covariant one?

Is there more than one type of multiplication WRITTEN IN TENSOR EQUATIONS?


Schwartz VANDSLIRE.:cool:
 

1. What is a covariant rank-2 tensor?

A covariant rank-2 tensor is a mathematical object that represents a linear transformation between two vector spaces. It is a combination of two covariant vectors and is denoted by a matrix.

2. What is a contravariant rank-2 tensor?

A contravariant rank-2 tensor is also a mathematical object that represents a linear transformation between two vector spaces. However, it is a combination of two contravariant vectors and is denoted by a matrix with superscript indices.

3. What is the difference between a covariant and contravariant rank-2 tensor?

The main difference between a covariant and contravariant rank-2 tensor lies in their transformation properties. A covariant tensor transforms in the same way as its underlying coordinate system, while a contravariant tensor transforms in the opposite way. This means that the components of a covariant tensor change when the coordinate system is changed, while the components of a contravariant tensor remain the same.

4. How are covariant and contravariant rank-2 tensors related?

Covariant and contravariant rank-2 tensors are related by the metric tensor, which is a mathematical object that defines the relationship between the two types of tensors. The metric tensor is used to convert between covariant and contravariant tensors, allowing for calculations to be performed in either type of tensor.

5. Where are covariant and contravariant rank-2 tensors used in science?

Covariant and contravariant rank-2 tensors are used in many fields of science, including physics, engineering, and mathematics. They are particularly useful in the study of relativity and electromagnetism, where they are used to describe the relationship between space and time, as well as the electromagnetic field. They are also used in fields such as fluid mechanics, continuum mechanics, and elasticity.

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