For the inhomogeneous anisotropic media these two forms of the diffusion equation should be combined in

For anisotropic materials the E and D fields are not parallel, and so E and D are related by the permittivity tensor ( a 2nd order tensor field ), in component form:

For the anisotropic material, it requires the mathematics of a second order tensor and up to 21 material property constants.

For an anisotropic material, the individual resistivity components, e. g. ρ < sub > x </ sub > or ρ < sub > y </ sub >, can be calculated using the Montgomery method.

For non-isotropic ( anisotropic ) coatings, the optimal coating depends on the shape of the target and the radar direction, but duality, the symmetry of Maxwell's equations between the electric and magnetic fields, tells one that optimal coatings have η < sub > 0 </ sub > × η < sub > 1 </ sub > = 377 < sup > 2 </ sup > Ω < sup > 2 </ sup >, where η < sub > 0 </ sub > and η < sub > 1 </ sub > are perpendicular components of the anisotropic surface impedance, aligned with edges and / or the radar direction.

For an anisotropic solid such as wood or paper, these three moduli do not contain enough information to describe its behaviour, and one must use the full generalized Hooke's law.

For anisotropic filtering, Radeon 8500 uses a technique similar to that used in R100, but improved with trilinear filtering and some other refinements.

For a time-independent interaction, i. e. heteronuclear dipolar couplings, CSA and first-order quadrupolar couplings, the anisotropic component is greatly reduced and almost suppressed in the limit of fast spinning, i. e. when the spinning frequency is greater than the breadth of the interaction.

For higher values, the surface diffusion of deposited atomic species would allow for the formation of crystallites with long range atomic order.

For an attractive interaction between particles, the diffusion coefficient tends to decrease as concentration increases.

For a repulsive interaction between particles, the diffusion coefficient tends to increase as concentration increases.

For fluorophores in solution this means that the intensity and polarization of the emitted light is dependent on rotational diffusion.

For biological molecules the diffusion coefficients normally range from 10 < sup >− 11 </ sup > to 10 < sup >− 10 </ sup > m < sup > 2 </ sup >/ s.

For the case of diffusion in two or more dimensions Fick's Second Law becomes

For more detail on diffusion length, see these examples.

For the diffusion equation this formula gives

For similar reasons, and also due to the small size of helium atoms, helium's diffusion rate through solids is three times that of air and around 65 % that of hydrogen.

For dilute gases, kinetic molecular theory relates the diffusion coefficient D to the particle density n = N / V, the molecular mass M, the collision cross section, and the absolute temperature T by

For example, gravitational convection can be seen in the diffusion of a source of dry salt downward into wet soil due to the buoyancy of fresh water in saline.

For another solute, bicarbonate, dialysis solution level is set at a slightly higher level than in normal blood, to encourage diffusion of bicarbonate into the blood, to act as a pH buffer to neutralize the metabolic acidosis that is often present in these patients.

For diffusion of heat ( thermal diffusion ), the Péclet number is defined as:

For diffusion of particles ( mass diffusion ), it is defined as:

For the latter portions of the process, boundary and lattice diffusion from the boundary become important.

For solids, on the other hand, diffusion of atoms to the surface might not be sufficient and the surface tension can vary with an increase in surface area.

For instance, the net rate at which a chemical dissolved in a fluid moves toward or away from some point is proportional to the Laplacian of the chemical concentration at that point ; expressed symbolically, the resulting equation is the diffusion equation.

For localized movement, cytoplasmic streaming within cells and slow diffusion of ions and molecules between cells are utilized.

For example, G841 is a dark blue made by Great American Market ( GAM ), and R02 is a light amber made by Rosco and L216 is a diffusion filter made by Lee Filters.

For a given amount of confusion and diffusion, an SP network has more " inherent parallelism "

For this reason path integrals were used in the study of Brownian motion and diffusion a while before they were introduced in quantum mechanics.

For a two-dimensional array, the element with indices i, j would have address B + c · i + d · j, where the coefficients c and d are the row and column address increments, respectively.

For instance, if all of the coefficients were equal to unity, a so-called boxcar function, then it would implement a low-pass filter with a low frequency gain of N + 1 and a frequency response given by the sinc function.

For similar reasons, if complex coefficients are allowed, one may have a single term like ( 2 + 3i ) x < sup > 3 </ sup >; even though it looks like it should be expanded to two terms, the complex number 2 + 3i is one complex number, and is the coefficient of that term.

For example, to represent the state of an n-qubit system on a classical computer would require the storage of 2 < sup > n </ sup > complex coefficients.

For one-dimensional simple harmonic motion, the equation of motion, which is a second-order linear ordinary differential equation with constant coefficients, could be obtained by means of Newton's second law and Hooke's law.

For each choice of coefficients, the resulting convex combination is a point in the convex hull, and the whole convex hull can be formed by choosing coefficients in all possible ways.

For a polynomial to be Hurwitz, it is necessary but not sufficient that all of its coefficients be positive.

For each one of these polynomials, there should be a best possible choice of coefficients a, b, c, and d that makes the approximation as good as possible.

For c, d, and higher-degree coefficients, these coefficients are determined by higher derivatives of f. c should always be f < nowiki ></ nowiki >( x < sub > 0 </ sub >)/ 2, and d should always be f < nowiki ></ nowiki >( x < sub > 0 </ sub >)/ 3 !.

For example, the coordinates of a vector in a vector space with basis, are the coefficients of the basis vectors in the expression

For example, in the formula, the number 2's in front of and are stoichiometric coefficients.

For higher degrees, the discriminant is always a polynomial function of the coefficients.

For instance, since is the ordinary generating function for binomial coefficients for a fixed n, one may ask for a bivariate generating function that generates the binomial coefficients for all k and n.

For example, the coefficients for BK7 above correspond to two absorption resonances in the ultraviolet, and one in the mid-infrared region.

For such structures the impulse response is equal to the filter coefficients.

For discrete filters the computational complexity is more or less proportional to the number of filter coefficients.

For larger matrices, the expressions for the coefficients c < sub > k </ sub > of the characteristic polynomial in terms of the matrix components become increasingly complicated ; but they can also be expressed in terms of traces of powers of the matrix A, using Newton's identities ( at least when the ring contains the rational numbers ), thus resulting in more compact expressions ( but which involve divisions by certain integers ).

For example, many of the authors of EGM96 are working on an updated model that should incorporate much of the new satellite gravity data ( see, e. g., GRACE ), and should support up to degree and order 2160 ( 1 / 6 of a degree, requiring over 4 million coefficients ).

For any field F, the ring of polynomials with coefficients in F is denoted by.

For algebraic purposes, the extension from rational numbers to real numbers is too " radical ": it introduces transcendental numbers, which are not the solutions of algebraic equations with rational coefficients.