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This shear is given by the off-diagonal elements of the stress tensor. It has recently been shown that the Maxwell stress tensor is the real part of a more general complex electromagnetic stress tensor whose imaginary part accounts for reactive electrodynamical forces.
This theory stipulated that all the laws of physics should take the same form in all coordinate systems – this led to the introduction of tensors. The tensor formalism also leads to a mathematically simpler presentation of physical laws. The inhomogeneous Maxwell equation leads to the continuity equation:
Let r(x) be the position vector of the point x with respect to the origin of the coordinate system. The notation can be simplified by noting that x = r(x). At each point we can construct a small line element dx. The square of the length of the line element is the scalar product dx • dx and is called the metric of the space.
In the mathematical description of general relativity, the Boyer–Lindquist coordinates [1] are a generalization of the coordinates used for the metric of a Schwarzschild black hole that can be used to express the metric of a Kerr black hole.
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.
Here are some formulas for conformal changes in tensors associated with the metric. (Quantities marked with a tilde will be associated with g ~ {\displaystyle {\tilde {g}}} , while those unmarked with such will be associated with g {\displaystyle g} .)
The sum of the entries along the main diagonal (the trace), plus one, equals 4 − 4(x 2 + y 2 + z 2), which is 4w 2. Thus we can write the trace itself as 2w 2 + 2w 2 − 1; and from the previous version of the matrix we see that the diagonal entries themselves have the same form: 2x 2 + 2w 2 − 1, 2y 2 + 2w 2 − 1, and 2z 2 + 2w 2 − 1. So ...
To derive the equation of the Mohr circle for the two-dimensional cases of plane stress and plane strain, first consider a two-dimensional infinitesimal material element around a material point (Figure 4), with a unit area in the direction parallel to the -plane, i.e., perpendicular to the page or screen.