Invariance of a tensor of order 2

  • #1
Thytanium
43
18
TL;DR Summary
In a tensor of order 1 (a vector) its magnitude, direction and sense must remain invariant given a change in coordinates; but in a tensor of order 2, I don't know what remains invariant.
Good morning friends of the Forum. For me it is difficult to geometrically imagine a tensor of order 2 and maybe that is why it is difficult for me to know, what remains invariant when making a change of coordinates of this tensor. The only thing I can think of it, is that since a tensor of order 2 is a tensor product of two vectors (Tensors of order 1), these two vectors remain invariant when changing coordinates and multiplying them tensorly. This I think is implicit in the definition of the order 2 tensor but I am not sure that these assertions I am making are correct. If you can clarify these doubts for me, I would appreciate it, friends of the Forum.
 
Physics news on Phys.org
  • #2
Thytanium said:
Good morning friends of the Forum.
Good evening!

Thytanium said:
For me it is difficult to geometrically imagine a tensor of order 2 and maybe that is why it is difficult for me to know, what remains invariant when making a change of coordinates of this tensor.
A tensor (including a vector) is a geometrical object. It is invariant by itself, what changes are the tensor components relative to whatever basis you choose to use.

Thytanium said:
The only thing I can think of it, is that since a tensor of order 2 is a tensor product of two vectors (Tensors of order 1), these two vectors remain invariant when changing coordinates and multiplying them tensorly.
More generally, a general tensor of order 2 is a linear combination of such products.
 
  • Like
Likes Kontilera
  • #3
Thanks you friend Orodruin. Thanhs you very much.
 

What is a tensor of order 2?

A tensor of order 2 is a mathematical object that can be represented as a matrix in a specific basis. It has two indices and transforms under a change of basis according to specific rules determined by its type (covariant, contravariant, or mixed). These tensors can represent various physical quantities, such as stress or strain in materials, or the metric tensor in spacetime.

What does it mean for a tensor of order 2 to be invariant?

Invariance of a tensor of order 2 means that the tensor remains unchanged under certain transformations, such as rotations or translations. This invariance is crucial in physics and engineering because it implies that the physical laws described by these tensors are the same in all coordinate systems or under all conditions encompassed by the transformation.

How can you determine if a tensor of order 2 is invariant under a transformation?

To determine if a tensor of order 2 is invariant under a specific transformation, you need to apply the transformation to each component of the tensor and check if the resulting tensor is equal to the original one. Mathematically, this involves using the transformation rules for tensors and verifying if the transformed components satisfy the condition T'ij = Tij for all i and j, where T'ij are the components of the transformed tensor and Tij are the components of the original tensor.

What are the implications of tensor invariance in physics?

Tensor invariance has significant implications in physics as it often corresponds to fundamental symmetries and conservation laws according to Noether's theorem. For example, the invariance of the metric tensor under time translations relates to the conservation of energy, while its invariance under spatial translations relates to the conservation of momentum. These principles are foundational in fields such as general relativity and quantum mechanics.

Can the invariance of a tensor change with different transformations?

Yes, the invariance of a tensor can change depending on the transformation applied. A tensor might be invariant under certain transformations (like rotations) but not under others (like non-orthogonal transformations). The specific nature of the tensor and the transformation involved determines this. Understanding these conditions is crucial for correctly applying tensors to describe physical systems and their symmetries.

Similar threads

I A couple questions about the Riemann Tensor, definition and convention
    • Differential Geometry
Replies
2
Views
2K
A Principal Invariants of the Weyl Tensor
    • Special and General Relativity
Replies
6
Views
1K
I How to derive irreducible representations/tensors?
    • Special and General Relativity
Replies
22
Views
2K
A Need some clarifications on tensor calculus please
    • Linear and Abstract Algebra
Replies
5
Views
1K
I Reconciling 2 expressions for Riemann curvature tensor
    • Differential Geometry
Replies
34
Views
2K
A On the dependence of the curvature tensor on the metric
    • Differential Geometry
Replies
6
Views
2K
I Basic Q about Vector/tensor definition and velocity
    • Differential Geometry
Replies
11
Views
2K
I Understanding tensor product and direct sum
    • Quantum Physics
Replies
11
Views
1K
A Geometrical interpretation of Ricci and Riemann tensors?
    • Differential Geometry
Replies
1
Views
5K
I What is the definition of trace for n-indexed tensor in group theory?
    • Linear and Abstract Algebra
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top