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It is therefore convenient to collect many of these terms in the Maxwell stress tensor, and to use tensor arithmetic to find the answer to the problem at hand. In the relativistic formulation of electromagnetism, the nine components of the Maxwell stress tensor appear, negated, as components of the electromagnetic stress–energy tensor , which ...
In continuum mechanics, including fluid dynamics, an upper-convected time derivative or Oldroyd derivative, named after James G. Oldroyd, is the rate of change of some tensor property of a small parcel of fluid that is written in the coordinate system rotating and stretching with the fluid.
In the fundamental branches of modern physics, namely general relativity and its widely applicable subset special relativity, as well as relativistic quantum mechanics and relativistic quantum field theory, the Lorentz transformation is the transformation rule under which all four-vectors and tensors containing physical quantities transform from one frame of reference to another.
The constitutive relations between the and F tensors, proposed by Minkowski for a linear materials (that is, E is proportional to D and B proportional to H), are: = = where u is the four-velocity of material, ε and μ are respectively the proper permittivity and permeability of the material (i.e. in rest frame of material), and denotes the ...
The process involves expressing () = (,) = (,,) in terms of x, y and z and replacing x, y and z with operators V x V y and V z which from vector operator. The resultant operator is hence a spherical tensor operator T ^ m ( l ) {\displaystyle {\hat {T}}_{m}^{(l)}} . ^ This may include constant due to normalization from spherical harmonics which ...
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 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]
More generally, if the Cartesian coordinates x, y, z undergo a linear transformation, then the numerical value of the density ρ must change by a factor of the reciprocal of the absolute value of the determinant of the coordinate transformation, so that the integral remains invariant, by the change of variables formula for integration.