Search results
Results from the WOW.Com Content Network
This discussion of tensors so far assumes finite dimensionality of the spaces involved, where the spaces of tensors obtained by each of these constructions are naturally isomorphic. [ Note 2 ] Constructions of spaces of tensors based on the tensor product and multilinear mappings can be generalized, essentially without modification, to vector ...
The wave function is encoded as a tensor contraction of a network of individual tensors. [5] The structure of the individual tensors can impose global symmetries on the wave function (such as antisymmetry under exchange of fermions) or restrict the wave function to specific quantum numbers, like total charge, angular momentum, or spin.
Cartesian tensors are widely used in various branches of continuum mechanics, such as fluid mechanics and elasticity. In classical continuum mechanics, the space of interest is usually 3-dimensional Euclidean space, as is the tangent space at each point.
libtensor [49] is a set of performance linear tensor algebra routines for large tensors found in post-Hartree–Fock methods in quantum chemistry. ITensor [50] features automatic contraction of matching tensor indices. It is written in C++ and has higher-level features for quantum physics algorithms based on tensor networks.
The Cauchy stress tensor is used for stress analysis of material bodies experiencing small deformations: it is a central concept in the linear theory of elasticity. For large deformations, also called finite deformations , other measures of stress are required, such as the Piola–Kirchhoff stress tensor , the Biot stress tensor , and the ...
The tensors and can be expressed by 3×3 matrices, relative to any chosen coordinate system. The fluid is said to be Newtonian if these matrices are related by the equation τ = μ ( ∇ v ) {\displaystyle {\boldsymbol {\tau }}={\boldsymbol {\mu }}(\nabla v)} where μ {\displaystyle \mu } is a fixed 3×3×3×3 fourth order tensor that does not ...
Some of the most common examples of transport analysis in engineering are seen in the fields of process, chemical, biological, [1] and mechanical engineering, but the subject is a fundamental component of the curriculum in all disciplines involved in any way with fluid mechanics, heat transfer, and mass transfer.
For infinitesimal deformations of a continuum body, in which the displacement gradient tensor (2nd order tensor) is small compared to unity, i.e. ‖ ‖, it is possible to perform a geometric linearization of any one of the finite strain tensors used in finite strain theory, e.g. the Lagrangian finite strain tensor, and the Eulerian finite strain tensor.