In light of the physical interpretation, a vector field with constant zero divergence is called incompressible or solenoidal – in this case, no net flow can occur across any closed surface.

One can express the divergence as a particular case of the exterior derivative, which takes a 2-form to a 3-form in R < sup > 3 </ sup >.

Thus, the divergence of the vector field F can be expressed as:

The divergence of a vector field can be defined in any number of dimensions.

Mutual information can be expressed as the average Kullback – Leibler divergence ( information gain ) of the posterior probability distribution of X given the value of Y to the prior distribution on X:

Or they can be launched into a beam of very low divergence in order to concentrate their power at a large distance.

The beam in the cavity and the output beam of the laser, when travelling in free space ( or a homogenous medium ) rather than waveguides ( as in an optical fiber laser ), can be approximated as a Gaussian beam in most lasers ; such beams exhibit the minimum divergence for a given diameter.

The divergence of a beam can be calculated if one knows the beam diameter at two separate points ( D < sub > i </ sub >, D < sub > f </ sub >), and the distance ( l ) between these points.

If the beam has been collimated using a lens or other focusing element, the divergence expected can be calculated from two parameters: the diameter,, of the narrowest point on the beam before the lens, and the focal length of the lens,.

It is usually used to describe connections between upper, middle or lower levels ( such as upper-level divergence causing lower level convergence in cyclone formation ), but can sometimes also be used to describe such connections over distance rather than height alone.

Also, one can take the divergence of any of these fluxes to determine the accumulation rate of the quantity in a control volume around a given point in space.

Sequence divergence in flanking regions can lead to poor primer annealing, especially at the 3 ’ section, where extension commences ; preferential amplification of particular size alleles due to the competitive nature of PCR can lead to heterozygous individuals being scored for homozygosity ( partial null ).

Letting Δ ( λ < sub > 0 </ sub >|| p ) denote the Kullback – Leibler divergence between an exponential with rate parameter λ < sub > 0 </ sub > and a predictive distribution p it can be shown that

The more general case is described by Poynting's theorem above, where it occurs as a divergence, which means that it can only describe the change of energy density in space, rather than the flow.

Since the Poynting vector only occurs in Poynting's theorem as a divergence ∇ • S, the Poynting vector S is arbitrary to the extent that one can add a curl of a field F to S,

In this case the flow can be determined completely from its kinematics: the assumptions of irrotationality and zero divergence of the flow.

Because the divergence of the magnetic field is always zero due to the absence of magnetic monopoles, such an A can always be found.

Vector fields can usefully be thought of as representing the velocity of a moving flow in space, and this physical intuition leads to notions such as the divergence ( which represents the rate of change of volume of a flow ) and curl ( which represents the rotation of a flow ).

The law can be expressed mathematically using vector calculus in integral form and differential form, both are equivalent since they are related by the divergence theorem, also called Gauss's theorem.

By the divergence theorem Gauss's law can alternatively be written in the differential form:

Whatever its location, this results in a divergence, with denser, nutrient-rich water being upwelled from below, and results in the remarkable fact that the equatorial region in the Pacific can be detected from space as a broad line of high phytoplankton concentration.

proposed a method that can generate a longitudinal electromagnetic ( light ) wave in free space, and this wave can propagate without divergence for a few wavelengths.

This may be because the Athabascan divergence began earlier ; ;

If the assets used are not identical ( so a price divergence makes the trade temporarily lose money ), or the margin treatment is not identical, and the trader is accordingly required to post margin ( faces a margin call ), the trader may run out of capital ( if they run out of cash and cannot borrow more ) and go bankrupt even though the trades may be expected to ultimately make money.

If the divergence is nonzero at some point then there must be a source or sink at that position.

If T is a ( p, q )- tensor ( p for the contravariant vector and q for the covariant one ), then we define the divergence of T to be the ( p, q − 1 )- tensor

By constructing a calibration curve of the ID of species ' pairs with known divergence times in the fossil record, the data could be used as a molecular clock to estimate the times of divergence of pairs with poorer or unknown fossil records.

Another interpretation of KL divergence is this: suppose a number X is about to be drawn randomly from a discrete set with probability distribution p ( x ).

Ethnicity, being largely developed by a divergence in geography, language, culture, genes and similarly, point of view, has the potential to be countered by a common source of information.

In evolution, the most important role of such chromosomal rearrangements may be to accelerate the divergence of a population into new species by making populations less likely to interbreed, and thereby preserving genetic differences between these populations.

The Cartan-derivative of the field form ( i. e. essentially the divergence of the field ) would be zero in the absence of the " gluon terms ", i. e. those ~ g, which represent the non-abelian character of the SU ( 3 ).

Covering the face is the subject of some divergence of opinion amongst the scholars – some consider it to be compulsory since the face is the major source of attraction, whilst others consider it to be highly recommended.

Conversely, Elijah Delmedigo ( c. 1458 – c. 1493 ), in his Bechinat ha-Dat endeavored to show that the Zohar could not be attributed to Shimon bar Yochai, arguing that if it were his work, the Zohar would have been mentioned by the Talmud, as has been the case with other works of the Talmudic period, that had bar Yochai known by divine revelation the hidden meaning of the precepts, his decisions on Jewish law from the Talmudic period would have been adopted by the Talmud, that it would not contain the names of rabbis who lived at a later period than that of Simeon ; and that if the Kabbalah was a revealed doctrine, there would have been no divergence of opinion among the Kabbalists concerning the mystic interpretation of the precepts.

If we choose the volume to be a ball of radius a around the source point, then Gauss ' divergence theorem implies that

A beam may, for example, have an elliptical cross section, in which case the orientation of the beam divergence must be specified, for example with respect to the major or minor axis of the elliptical cross section.

Gaussian laser beams are said to be diffraction limited when their radial beam divergence is close to the minimum possible value, which is given by

Unlike the gradient and divergence, curl does not generalize as simply to other dimensions ; some generalizations are possible, but only in three dimensions is the geometrically defined curl of a vector field again a vector field.

More rigorously, the divergence of a vector field F at a point p is defined as the limit of the net flow of F across the smooth boundary of a three dimensional region V divided by the volume of V as V shrinks to p. Formally,

If a vector field F with zero divergence is defined on a ball in R < sup > 3 </ sup >, then there exists some vector field G on the ball with F = curl ( G ).

The divergence is then the function defined by

When applied to a field ( a function defined on a multi-dimensional domain ), del may denote the gradient ( locally steepest slope ) of a scalar field ( or sometimes of a vector field, as in the Navier – Stokes equations ), the divergence of a vector field, or the curl ( rotation ) of a vector field, depending on the way it is applied.

The Laplace operator is a second order differential operator in the n-dimensional Euclidean space, defined as the divergence (∇·) of the gradient (∇ ƒ ).

From a mathematical point of view the IR divergences can be regularized by assuming fractional differentiation with respect to a parameter, for example is well defined at p = a but is UV divergent, if we take the 3 / 2-th fractional derivative with respect to we obtain the IR divergence, so we can cure IR divergences by turning them into UV divergences.

The divergence of a congruence is defined

* Version 2: As long as private property rights are well defined under zero transaction cost, exchange will eliminate divergence and lead to efficient use of resources or highest valued use of resources.

Vector operators are defined in terms of del, and include the gradient, divergence, and curl:

their K – L divergence is defined to be

In words, it is the average of the logarithmic difference between the probabilities P and Q, where the average is taken using the probabilities P. The K-L divergence is only defined if P and Q both sum to 1 and if for any i such that.

divergence from P to Q is defined as

The ( super -) divergence of a vector field is defined as

Up to a sign factor, it is defined as one half the divergence of the corresponding Hamiltonian vector field,

The formula above holds for surfaces in 3D space defined in any manner, as long as the divergence of the unit normal may be calculated.

This identity is derived from the divergence theorem applied to the vector field: Let φ and ψ be scalar functions defined on some region U in R < sup > 3 </ sup >, and suppose that φ is twice continuously differentiable, and ψ is once continuously differentiable.

In general relativity, the four-current is defined as the divergence of the electromagnetic displacement, defined as

As well as the absolute Rényi entropies, Rényi also defined a spectrum of divergence measures generalising the Kullback – Leibler divergence.

The Rényi divergence of order α, where α > 0, from a distribution P to a distribution Q is defined to be:

Like the Laplacian, the Laplace – Beltrami operator is defined as the divergence of the gradient, and is a linear operator taking functions into functions.