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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 principal invariants of the Riemann and Weyl tensors are certain quadratic polynomial invariants (i.e., sums of squares of components). The principal invariants of the Riemann tensor of a four-dimensional Lorentzian manifold are the Kretschmann scalar =
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.
The principal invariants of tensors do not change with rotation of the coordinate system (see Invariants of tensors). The singular values of a matrix are invariant under orthogonal transformations. Lebesgue measure is invariant under translations. The variance of a probability distribution is invariant under translations of the real line.
[1] are a set of tensor invariants that span the space of real, symmetric, second-order, 3-dimensional tensors and are isomorphic with respect to principal stress space. This right-handed orthogonal coordinate system is named in honor of the German scientist Dr. Walter Lode because of his seminal paper written in 1926 describing the effect of ...
The derivatives of scalars, vectors, and second-order tensors with respect to second-order tensors are of considerable use in continuum mechanics. These derivatives are used in the theories of nonlinear elasticity and plasticity, particularly in the design of algorithms for numerical simulations. [1]
Today's NYT Connections puzzle for Friday, December 13, 2024The New York Times
The vectors appearing in this minimal expression are the principal axes of the tensor, and generally have an important physical meaning. For example, the principal axes of the inertia tensor define the Poinsot's ellipsoid representing the moment of inertia. Also see Sylvester's law of inertia. For symmetric tensors of arbitrary order k ...