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A symmetric tensor is a special type of tensor that exhibits a distinct behavior with respect to its indices.
By definition, a tensor is called symmetric when swapping any pair of its indices doesn’t change its value.
For a rank-two tensor, like a matrix, if we have the elements arranged as \( T^{ij} \) and swapping these indices \( T^{ij} = T^{ji} \), the tensor is symmetric.
In simpler terms, the element at position \((i, j)\) is equal to the element at position \((j, i)\).

• A symmetric tensor will have its elements mirrored across the main diagonal.
• This property makes symmetric tensors particularly useful in physics and engineering as they can represent quantities like stress and strain, where directionality is mirrored.

A symmetric tensor can be decomposed from any general tensor by taking the average of each pair of entries and its swapped counterpart.
This decomposition is expressed as: \( S^{ij} = \frac{1}{2}(T^{ij} + T^{ji}) \).
This ensures that the resulting tensor retains its symmetry across the diagonal.

Antisymmetric tensors, conversely, have a property that starkly contrasts with symmetric tensors.
An antisymmetric tensor changes sign whenever two of its indices are swapped.
For a rank-two tensor: \( T^{ij} = -T^{ji} \).

• If you exchange any pair of indices in an antisymmetric tensor, the value becomes negative.
• The diagonal elements of an antisymmetric tensor are always zero because \( T^{ii} = -T^{ii} \), and the only number equal to its own negative is zero.

The antisymmetric part of a general tensor \( T^{ij} \), can be derived by the formula: \( A^{ij} = \frac{1}{2}(T^{ij} - T^{ji}) \).
This expression ensures that the resulting tensor changes sign with an index swap, preserving antisymmetry.
Antisymmetric tensors are essential in physics, particularly in electromagnetism and fluid mechanics, where they represent rotational characteristics.

Tensors are versatile mathematical entities used to generalize scalar, vector, and more complex relations across spaces.
A contravariant tensor, specifically, transforms a certain way under coordinate transformations.

• In simple terms, the components of a contravariant tensor change in opposition to the change in the coordinate grid. This means if the coordinate grid scales up, the tensor's components scale down, and vice versa.
• Mathematically, if \( x^i \) are the original coordinates and \( \bar{x}^i \) are the new coordinates, the components of a contravariant tensor transform as: \( T^{i'} = \frac{\partial x^{i'}}{\partial x^i} T^i \).

Understanding contravariant tensors is crucial when dealing with spaces where direction and magnitude both matter, such as in physics and advanced calculus.
They lay the foundation for more profound operations and transformations, essential when tensors interact with complex spaces and systems.
Whether decomposing into symmetric and antisymmetric components or transforming across coordinate grids, contravariant tensors maintain their fundamental properties throughout.