- #1
arpon
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Is that true in general and why:
$$A^{mn}_{.~.~lm}=A^{nm}_{.~.~ml}$$
$$A^{mn}_{.~.~lm}=A^{nm}_{.~.~ml}$$
That's the requirement thathaushofer said:It's only true when your tensor is symmetric in both upper and lower indices.
A mixed tensor is a mathematical object that represents a linear mapping between two vector spaces. It is a combination of both covariant and contravariant components, which can be thought of as representing different directions in space.
A mixed tensor is contracted by summing over one covariant and one contravariant index, resulting in a scalar value. This is known as the Einstein summation convention.
Contracting a mixed tensor allows us to extract useful information and simplify calculations. It can help us determine if two tensors are equivalent, find the eigenvalues of a tensor, or represent physical quantities such as stress or strain.
Contraction of tensors involves summing over specific indices, while multiplication involves multiplying all corresponding components. Contraction results in a scalar value, while multiplication results in a tensor of higher rank.
The contraction of a mixed tensor is closely related to the trace of a matrix. In fact, the trace can be seen as a special case of contraction, where the tensor has equal covariant and contravariant components. The trace represents the sum of the diagonal elements of a matrix, while contraction can involve any indices.