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BobbyFluffyPric
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Is the matrix of a second order symmetric tensor always symmetric (ie. expressed in any coordinate system, and in any basis of the coordinate system)?
Please help!
~Bee
Please help!
~Bee
BobbyFluffyPric said:Is the matrix of a second order symmetric tensor always symmetric (ie. expressed in any coordinate system, and in any basis of the coordinate system)?
Please help!
~Bee
BobbyFluffyPric said:Explain, thank you, I'm more assured now , even though I don't understand the thing about thinking of "contravariant vectors and covariant vectors as things coming from two different vector spaces" - as far as I know, a vector is a vector, and one can only talk of contravariant or covariant components of the same (the reason we distinguish between the contravariant and covariant basis vectors is just a question of nomenclature, but both sets are no more than specific sets of vectors in the same vector space). But maybe that is another issue .
~Bee
Have you seen the difference between a "vector" and an "axial vector" / "pseudovector"? That is very closely related to the difference between a "contravariant vector" and a "covariant vector".BobbyFluffyPric said:It's hard for me not to think of a vector, which is a geometrical entity that one can represent via an arrow (eg force, velocity etc)
BobbyFluffyPric said:Explain, I wonder if you can recommend me some text that explains this in detail, although I suppose that as an engineering student I shall have little need to delve further into it. However, I am curious as to this distinction between contravariant and covariant vectors (I actually read in some maths text that though the terms contravariant vector and covariant vector were being used, strictly speaking these terms should be applied to the components of the a vector, and thus I took it for granted that this was the case in all texts that referred to vectors as covariant or contravariant). It's hard for me not to think of a vector, which is a geometrical entity that one can represent via an arrow (eg force, velocity etc), as not being the same physical entity regardless of what type of components are used to describe it (ie regardless of the basis it is expressed in).
A symmetric tensor is a multidimensional array of numbers that has the property of being invariant under certain transformations. In other words, it remains unchanged when the order of its indices is permuted. A symmetric matrix, on the other hand, is a square matrix that is equal to its own transpose. This means that the elements above the main diagonal are the same as the elements below the main diagonal.
Symmetric tensors and matrices are commonly used in physics to represent physical quantities that are independent of the coordinate system used to describe them. For example, stress and strain tensors are symmetric because they are the same regardless of the direction of measurement. These tensors and matrices also play a crucial role in describing the symmetries of physical systems, such as in the study of crystal structures.
Yes, symmetric tensors and matrices are commonly used in machine learning algorithms, especially in tasks such as image and signal processing. They can be used to represent data that has inherent symmetry, such as images and audio signals. This allows for more efficient and accurate processing of the data.
A tensor is considered symmetric if it remains unchanged under a permutation of its indices. This means that if the order of the indices is changed, the values of the tensor remain the same. For a matrix to be symmetric, it must be equal to its own transpose, meaning that the elements above the main diagonal are the same as the elements below the main diagonal.
Yes, symmetric tensors and matrices have various applications in engineering, particularly in the fields of mechanics and structural engineering. They are used to represent physical quantities, such as stress and strain, in structures and materials. They also play a crucial role in analyzing the symmetries of structures, which can help in designing more efficient and stable systems.