The idea was to show that a tensor with symmetric components in one coordinate system has symmetric components in all coordinate systems. Above this meand that the exchange of p and q in ##T_{pq}## is assumed to be fine and we want to show that this implies that i and j can be exchanged. The OP's question was regarding the validity of changing the order of the transformation coefficients.jedishrfu said:Since the tensor is symmetric then you can interchange the i and j in the expression.
Right?
A symmetric tensor is a mathematical object that represents a physical quantity or property that remains unchanged when the coordinate system is rotated or reflected. It is a type of tensor that is characterized by having equal values for certain components that are related by symmetry.
Invariance refers to the property of remaining unchanged under certain transformations. In the case of symmetric tensors, invariance means that the tensor remains the same when the coordinate system is rotated or reflected, regardless of the specific values of the components.
Symmetric tensors have a wide range of applications in physics, engineering, and other fields. They are commonly used in mechanics, electromagnetism, and fluid dynamics to represent physical quantities such as stress, strain, and electric fields. They are also used in image and signal processing, as well as in materials science and computer graphics.
To determine if a tensor is symmetric, you can use the symmetry conditions for each component. For a second-order symmetric tensor, the condition is that the components must be equal when the indices are swapped. For higher-order tensors, there are more conditions to check, but the principle remains the same.
Yes, symmetric tensors can often be used to simplify calculations, since they have fewer independent components than general tensors. This allows for a more efficient representation of physical quantities and can lead to simpler and more elegant solutions to problems.