What is the physical interpretation of the trace of a

We have introduced two 3-scalar fields (x, ) and (x, ), one 3-vector field w(x, ) = w i e i, and one symmetric, traceless second-rank 3-tensor field h(x, ) = h ij e i e j. No generality is lost by making h ij traceless since any trace part can be put into . The factors of 2 and signs have been chosen to simplify later expressions. Aug 22, 2019 · We prove rigorously that the symmetric traceless and the antisymmetric tensor models in rank three with tetrahedral interaction admit a 1 / N expansion, and that at leading order they are dominated by melon diagrams. This proves the recent conjecture of Klebanov and Tarnopolsky (JHEP 10:037, 2017. arXiv:1706.00839 ), which they checked numerically up to 8th order in the coupling As for the energy-momentum tensor, in general it is neither symmetric nor traceless. In a Poincare symmetric theory, the energy-momentum tensor can be made symmetric. And, in order to make it symmetric and traceless, the theory must be invariant under a larger space-time symmetry group called the Conformal group. An interesting aspect of a traceless tensor is that it can be formed entirely from shear components. For example, a coordinate system transformation can be found to express the deviatoric strain tensor in the above example as shear strain exclusively. In the screenshot here, the above deviatoric strain tensor was input into the webpage, and symmetric, traceless tensor with 5 independent terms. Meanwhile, suppose that T = x ip j where xand pare position and momentum operators. The tensor will include a scalar x p a vector x p and a symmetric traceless tensor 1 2 (x ip j+ x jp i) 1 3 x p. We could write the operator in a spherical basis using spherical harmonics where we let Y 1 1

## Third order three-dimensional symmetric and traceless tensors play an important role in physics and tensor representation theory. A minimal integrity basis of a third order three-dimensional symmetric and traceless tensor has four invariants with degrees two, four, six, and ten, respectively. In this paper, we show that any minimal integrity basis of a third order three-dimensional symmetric

Scalar-vector-tensor decomposition - Wikipedia

### As the gravity gradient tensor Ψ is symmetric and is traceless, it has five independent quantities to interpret. Interpreting two quantities: the invariants, the eigenvalues, the modulus and the phase, or the modulus and the shape index cannot give a comp

Jul 19, 2020 · A zero rank tensor is a scalar, a first rank tensor is a vector; a one-dimensional array of numbers. A second rank tensor looks like a typical square matrix. Stress, strain, thermal conductivity, magnetic susceptibility and electrical permittivity are all second rank tensors. An interesting aspect of a traceless tensor is that it can be formed entirely from shear components. For example, a coordinate system transformation can be found to express the deviatoric stress tensor in the above example as shear stress exclusively. In the screenshot here, the above deviatoric stress tensor was input into the webpage, and Transverse, traceless, symmetric and not forget $(\square-m^2)$. In fact, it need not be symmetric, it can be any irreducible tensor of the Lorentz algebra (symmetric and traceless is just the simplest type of irreducible tensor). on the use of the traceless stress tensor (TST). It is shown that it naturally leads to the appearance of a modiﬁed viscosity given by C. =3/ tr.˝/ where is the shear-viscosity coefﬁcient, the relaxation time and tr(˝) the trace of the extra stress tensor. This modiﬁed viscosity reaches high values near singular points, the troublesome The first term, the dot product of the two vectors, is clearly a scalar under rotation, the second term, which is an antisymmetric tensor has three independent components which are the vector components of the vector product U → × V →, and the third term is a symmetric traceless tensor, which has five independent components.