Is this Matrix a Tensor, Pseudotensor, or Neither?

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In summary, a tensor is a mathematical object that describes the relationship between different coordinate systems in a geometric space, while a pseudotensor is a quantity that has tensor-like properties but does not transform like a tensor under certain coordinate transformations. Tensors have a clear physical interpretation, while pseudotensors can be more complex and less intuitive. Tensors and pseudotensors cannot be converted into each other, but pseudotensors can be constructed from tensors. Tensors are more useful in applications where physical interpretation is important, while pseudotensors are more useful in applications where symmetry is important. Both tensors and pseudotensors are used in different branches of science, with tensors being more commonly used in physics and engineering and pseudotensors being more commonly used
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Marthius
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I am having trouble understanding how to tell if a matrix is a tensor, a pseudo tensor or neither or the above.
Given this matrix
[tex]\begin{vmatrix}2x^-2y^2 & 4xy\\ 4xy & -2x^2+2y^2\end{vmatrix}[/tex]
I thought of trying just inverting the components and checking the sign, but I don't think that this will tell me if it is actually a tensor.
 
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  • #2
Can you give your definition of 'tensor' and 'pseudo tensor'?
 
  • #3
Actualy I don't think I can, that may be part of the issue.
 

1. What is the difference between a tensor and a pseudotensor?

A tensor is a mathematical object that describes the relationship between different coordinate systems in a geometric space. It has components that transform in a specific way under coordinate transformations. A pseudotensor, on the other hand, is a mathematical object that transforms like a tensor under certain coordinate transformations, but not under others. This means that it is not a true tensor, but rather a quantity that has tensor-like properties.

2. How do tensors and pseudotensors differ in terms of their physical interpretations?

Tensors have a clear physical interpretation in terms of their components representing quantities such as displacement, velocity, and stress. Pseudotensors, on the other hand, do not have such a clear physical interpretation. They can represent things like angular momentum, but their interpretation can be more complex and less intuitive.

3. Can a tensor be converted into a pseudotensor or vice versa?

No, a tensor cannot be converted into a pseudotensor or vice versa. Tensors and pseudotensors are fundamentally different mathematical objects and cannot be transformed into one another. However, it is possible to construct a pseudotensor from a tensor by introducing additional factors, but this does not change the fact that they are fundamentally different objects.

4. Are there specific applications where tensors are more useful than pseudotensors?

Yes, tensors are more useful in applications where the physical interpretation of the mathematical object is important. This includes areas such as fluid mechanics, electromagnetism, and general relativity. Pseudotensors, on the other hand, are more useful in applications where the symmetry of the mathematical object is important, such as in crystallography and solid state physics.

5. Are tensors and pseudotensors used in different branches of science?

Yes, tensors and pseudotensors are used in different branches of science. Tensors are commonly used in physics and engineering, while pseudotensors are more commonly used in mathematics and crystallography. However, they can both be used in a variety of fields depending on the specific application.

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