The divergence of a continuously differentiable second-rank tensor field is a first-rank tensor field:

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.

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.

In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar.

More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.

In physical terms, the divergence of a three dimensional vector field is the extent to which the vector field flow behaves like a source or a sink at a given point.

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,

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.

where e < sub > a </ sub > is the unit vector in direction a, the divergence is

The divergence of the curl of any vector field ( in three dimensions ) is equal to zero:

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 ).

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

Note however that working with the current two form itself and the exterior derivative is usually easier than working with the vector field and divergence, because unlike the divergence, the exterior derivative commutes with a change of ( curvilinear ) coordinate system.

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

In Einstein notation, the divergence of a contravariant vector is given by

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

* The idea of divergence of a vector field

* Div ( divergence ) is a vector operator that measures a vector field's divergence from or convergence towards a given point.

This is in analogy to electrostatics, in which the electric field ( E-field ) has a vanishing curl and the magnetic field ( B-field ) has a vanishing divergence.

Owing respectively to Green's theorem and the divergence theorem, such a field is necessarily conserved and free from sources or sinks, having net flux equal to zero through any open domain.

Therefore the divergence of the velocity field in that region would have a positive value, as the region is a source.

The Laplacian of a scalar field is the divergence of the field's gradient:

With this additional condition — the covariant divergence of the energy – momentum tensor, and hence of whatever is on the other side of the equation, is zero — the simplest set of equations are what are called Einstein's ( field ) equations:

where c is the speed of light and ∇· denotes the divergence of a tensor field.

The common notation for the divergence ∇· F is a convenient mnemonic, where the dot denotes an operation reminiscent of the dot product: take the components of ∇ ( see del ), apply them to the components of F, and sum the results.

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,

since the divergence of the curl term is zero: ∇ • (∇ × F ) = 0 for an arbitrary field F ( see Vector calculus identities ).

* Applying the divergence theorem to the product of a scalar function g and a vector field F, the result is

* Applying the divergence theorem to the cross-product of a vector field F and a non-zero constant vector, the following theorem can be proven:

The main point of divergence in this timeline was the 1960 Presidential election-here, John F. Kennedy was caught in an affair with Marilyn Monroe, crippling his support and giving Richard Nixon the Presidency.

The point of divergence for this alternate timeline happens on November 22, 1963, where John F. Kennedy survived the assassination ( Jacqueline Kennedy was killed, in the renaming of the Kennedy Space Center as the Jacqueline B. Kennedy Space Center ), but was crippled and thus incapacitated, as Lyndon B. Johnson is still sworn in.

* C. Colliard, A. Sicilia, G. F. Turrisi, M. Arculeo, N. Perrin and M. Stöck: Strong reproductive barriers in a narrow hybrid zone of West-Mediterranean green toads ( Bufo viridis subgroup ) with Plio-Pleistocene divergence.

where ∇ ⋅ = div is the divergence, and ∇ = grad is the gradient.

Differentiating the above divergence with respect to ε at ε = 0 and changing the sign yields the conservation law

If magnetic charges do not exist – or if they do exist but are not present in a region of space – then the new terms in Maxwell's equations are all zero, and the extended equations reduce to the conventional equations of electromagnetism such as ∇• B = 0 ( where ∇• is divergence and B is the magnetic B field ).

where is the divergence operator, D = electric displacement field, and ρ < sub > f </ sub > = free charge density ( describing charges brought from outside ).

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 decay is caused by the decrease in solar insolation, divergence of turbulent flux and relaxation of lateral gradients .< ref name = " Caldwell ">

1, which gives the Shannon entropy and the Kullback – Leibler divergence, is special because it is only when α = 1 that one can separate out variables A and X from a joint probability distribution, and write:

In particular, the information gain about a random variable X obtained from an observation that a random variable A takes the value A = a is the Kullback-Leibler divergence D < sub > KL </ sub >( p ( x | a ) || p ( x | I ) ) of the prior distribution p ( x | I ) for x from the posterior distribution p ( x | a ) for x given a.

( where the first map is the gradient, the second is the curl, the third is the divergence ) serves as a nice quantification of the complicatedness of the underlying region U. These are the beginnings and main motivations of de Rham cohomology.

The U. S. Air Force also participated in the program, but different requirements led to some divergence in the development of NASA and USAF Scouts.

The divergence between east coast hip hop and west coast hip hop was soon imported into Romania with the creation of the Cartel (" Cartelul ") representing the west coast with groups like B. U. G.

For example, Jochen Hartwig provides evidence to show that " the divergence in growth rates real GDP between the U. S. and the EU since 1997 can be explained almost entirely in terms of changes to deflation methods that have been introduced in the U. S. after 1997, but not-or only to a very limited extent-in Europe ".