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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.
For two-dimensional, plane strain problems the strain-displacement relations are = ; = [+] ; = Repeated differentiation of these relations, in order to remove the displacements and , gives us the two-dimensional compatibility condition for strains
The first line in the action contains the Einstein–Hilbert action, the dilaton kinetic term, and the Kalb–Ramond field strength tensor =. The first term on the second line has the appropriately modified field strength tensors for the three C p {\displaystyle C_{p}} gauge fields, while the last term is a Chern–Simons term .
The tensors are classified according to their type (n, m), where n is the number of contravariant indices, m is the number of covariant indices, and n + m gives the total order of the tensor. For example, a bilinear form is the same thing as a (0, 2) -tensor; an inner product is an example of a (0, 2) -tensor, but not all (0, 2) -tensors are ...
Totally antisymmetric tensors include: Trivially, all scalars and vectors (tensors of order 0 and 1) are totally antisymmetric (as well as being totally symmetric). The electromagnetic tensor, in electromagnetism. The Riemannian volume form on a pseudo-Riemannian manifold.
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.
It is the viscous stress that occurs in fluid moving through a tube with uniform cross-section (a Poiseuille flow) or between two parallel moving plates (a Couette flow), and resists those motions. The part E v of E acts as a scalar multiplier (like ε v), the average expansion rate of the medium around the point in question. (It is represented ...
The space of symmetric tensors of order r on a finite-dimensional vector space V is naturally isomorphic to the dual of the space of homogeneous polynomials of degree r on V. Over fields of characteristic zero, the graded vector space of all symmetric tensors can be naturally identified with the symmetric algebra on V.