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A real tensor in 3D (i.e., one with a 3x3 component matrix) has as many as six independent invariants, three being the invariants of its symmetric part and three characterizing the orientation of the axial vector of the skew-symmetric part relative to the principal directions of the symmetric part.
The vector spaces of a tensor product need not be the same, and sometimes the elements of such a more general tensor product are called "tensors". For example, an element of the tensor product space V ⊗ W is a second-order "tensor" in this more general sense, [29] and an order-d tensor may likewise be defined as an element of a tensor product ...
In theoretical physics, an invariant is an observable of a physical system which remains unchanged under some transformation. Invariance, as a broader term, also applies to the no change of form of physical laws under a transformation, and is closer in scope to the mathematical definition .
There are many examples of symmetric tensors. Some include, the metric tensor, , the Einstein tensor, and the Ricci tensor, .. Many material properties and fields used in physics and engineering can be represented as symmetric tensor fields; for example: stress, strain, and anisotropic conductivity.
Second degree examples are called quadratic invariants, and so forth. Invariants constructed using covariant derivatives up to order n are called n-th order differential invariants . The Riemann tensor is a multilinear operator of fourth rank acting on tangent vectors .
The second fundamental form of a parametric surface S in R 3 was introduced and studied by Gauss. First suppose that the surface is the graph of a twice continuously differentiable function, z = f ( x , y ) , and that the plane z = 0 is tangent to the surface at the origin.
Similarly, every second rank tensor (such as the stress and the strain tensors) has three independent invariant quantities associated with it. One set of such invariants are the principal stresses of the stress tensor, which are just the eigenvalues of the stress tensor. Their direction vectors are the principal directions or eigenvectors.
Another possible invariant (which has been employed for example in writing the gravitational term of the Lagrangian for some higher-order gravity theories) is . where is the Weyl tensor, the conformal curvature tensor which is also the completely traceless part of the Riemann tensor.