In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the rate that the vector field alters the volume in an infinitesimal neighborhood of each point. (In 2D this "volume" refers to area.) More precisely, the divergence at a point is the rate that the flow of the vector field modifies a volume about the point in the limit, as a small volume shrinks down to the point.
As an example, consider air as it is heated or cooled. The velocity of the air at each point defines a vector field. While air is heated in a region, it expands in all directions, and thus the velocity field points outward from that region. The divergence of the velocity field in that region would thus have a positive value. While the air is cooled and thus contracting, the divergence of the velocity has a negative value.
Physical interpretation of divergence
In physical terms, the divergence of a vector field is the extent to which the vector field flux behaves like a source or a sink at a given point. It is a local measure of its "outgoingness" – the extent to which there are more of the field vectors exiting from an infinitesimal region of space than entering it. A point at which the flux is outgoing has positive divergence, and is often called a "source" of the field. A point at which the flux is directed inward has negative divergence, and is often called a "sink" of the field. The greater the flux of field through a small surface enclosing a given point, the greater the value of divergence at that point. A point at which there is zero flux through an enclosing surface has zero divergence.
The divergence of a vector field is often illustrated using the simple example of the velocity field of a fluid, a liquid or gas. A moving gas has a velocity, a speed and direction at each point, which can be represented by a vector, so the velocity of the gas forms a vector field. If a gas is heated, it will expand. This will cause a net motion of gas particles outward in all directions. Any closed surface in the gas will enclose gas which is expanding, so there will be an outward flux of gas through the surface. So the velocity field will have positive divergence everywhere. Similarly, if the gas is cooled, it will contract. There will be more room for gas particles in any volume, so the external pressure of the fluid will cause a net flow of gas volume inward through any closed surface. Therefore, the velocity field has negative divergence everywhere. In contrast, in a gas at a constant temperature and pressure, the net flux of gas out of any closed surface is zero. The gas may be moving, but the volume rate of gas flowing into any closed surface must equal the volume rate flowing out, so the net flux is zero. Thus the gas velocity has zero divergence everywhere. A field which has zero divergence everywhere is called solenoidal.
If the gas is heated only at one point or small region, or a small tube is introduced which supplies a source of additional gas at one point, the gas there will expand, pushing fluid particles around it outward in all directions. This will cause an outward velocity field throughout the gas, centered on the heated point. Any closed surface enclosing the heated point will have a flux of gas particles passing out of it, so there is positive divergence at that point. However any closed surface not enclosing the point will have a constant density of gas inside, so just as many fluid particles are entering as leaving the volume, thus the net flux out of the volume is zero. Therefore, the divergence at any other point is zero.
Definition
The divergence of a vector field at a point is defined as the limit of the ratio of the surface integral of out of the closed surface of a volume enclosing to the volume of , as shrinks to zero
where is the volume of , is the boundary of , and is the outward unit normal to that surface. It can be shown that the above limit always converges to the same value for any sequence of volumes that contain and approach zero volume. The result, , is a scalar function of .
Since this definition is coordinate-free, it shows that the divergence is the same in any coordinate system. However the above definition is not often used practically to calculate divergence; when the vector field is given in a coordinate system the coordinate definitions below are much simpler to use.
A vector field with zero divergence everywhere is called solenoidal – in which case any closed surface has no net flux across it. This is the same as saying that the (flow of the) vector field preserves volume: The volume of any region does not change after it has been transported by the flow for any period of time.
Definition in coordinates Cartesian coordinates
In three-dimensional Cartesian coordinates, the divergence of a continuously differentiable vector field is defined as the scalar-valued function:
Although expressed in terms of coordinates, the result is invariant under rotations, as the physical interpretation suggests. This is because the trace of the Jacobian matrix of an -dimensional vector field in -dimensional space is invariant under any invertible linear transformation.
The common notation for the divergence () is a convenient mnemonic, where the dot denotes an operation reminiscent of the dot product: take the components of the operator (see del), apply them to the corresponding components of , and sum the results. Because applying an operator is different from multiplying the components, this is considered an abuse of notation.
Cylindrical coordinates For a vector expressed in local unit cylindrical coordinates as
where is the unit vector in direction , the divergence is
The use of local coordinates is vital for the validity of the expression. If we consider the position vector and the functions , , and , which assign the corresponding global cylindrical coordinate to a vector, in general and In particular, if we consider the identity function , we find that:
Spherical coordinates In spherical coordinates, with the angle with the axis and the rotation around the axis, and again written in local unit coordinates, the divergence is
Tensor field Let be continuously differentiable second-order tensor field defined as follows:
The divergence of a second-order tensor represents a contraction, and the result is a first-order tensor field (i.e., classically, a vector). Somewhat unfortunately, for Cartesian coordinates the operation has been routinely defined in two different ways. The first is given by (noting there is no sum over the repeated index of the basis vectors)
and the second (more common) form by
The second of these was the form suggested by the progenitor of vector algebra, J. Willard Gibbs; thus if one is following Gibbsian notation strictly, this should be the form adopted. Note that if a tensor is symmetric then the two forms are equivalent. Because of this, often in the literature the two definitions (and symbols and ) are used interchangeably (especially in mechanics equations where tensor symmetry is usually assumed).
Expressions of in cylindrical and spherical coordinates are given in the article del in cylindrical and spherical coordinates; that article adopts the second of the two definitions above.
General coordinates Using Einstein notation we can consider the divergence in general coordinates, which we write as , where is the number of dimensions of the domain. Here, the upper index refers to the number of the coordinate or component, so refers to the second component, and not the quantity squared. The index variable is used to refer to an arbitrary component, such as . The Voss-Weyl formula, which allows the divergence to be determined using simply partial coordinate derivatives, is as follows:
where is the local coefficient of the volume element and are the components of with respect to the local unnormalized covariant basis (sometimes written as The Einstein notation implies summation over , since it appears as both an upper and lower index.
The volume coefficient is a function of position which depends on the coordinate system. In Cartesian, cylindrical and spherical coordinates, using the same conventions as before, we have , and , respectively. The volume can also be expressed as , where is the metric tensor. The determinant appears because it provides the appropriate invariant definition of the volume, given a set of vectors. Since the determinant is a scalar quantity which doesn't depend on the indices, these can be suppressed, writing The absolute value is taken in order to handle the general case where the determinant might be negative, such as in pseudo-Riemannian spaces. The reason for the square-root is a bit subtle: it effectively avoids double-counting as one goes from curved to Cartesian coordinates, and back. The volume (the determinant) can also be understood as the Jacobian of the transformation from Cartesian to curvilinear coordinates, which for gives
Some conventions expect all local basis elements to be normalized to unit length, as was done in the previous sections. If we write for the normalized basis, and for the components of with respect to it, we have that
using one of the properties of the metric tensor. By dotting both sides of the last equality with the contravariant element we can conclude that . After substituting, the formula becomes:
See for further discussion.
Properties
The following properties can all be derived from the ordinary differentiation rules of calculus. Most importantly, the divergence is a linear operator, i.e.,
for all vector fields and and all real numbers and .
There is a product rule of the following type: if is a scalar-valued function and is a vector field, then
or in more suggestive notation
Another product rule for the cross product of two vector fields and in three dimensions involves the curl and reads as follows:
or
The Laplacian of a scalar field is the divergence of the field's gradient:
The divergence of the curl of any vector field (in three dimensions) is equal to zero:
If a vector field with zero divergence is defined on a ball in , then there exists some vector field on the ball with . For regions in more topologically complicated than this, the latter statement might be false (see Poincaré lemma). The degree of failure of the truth of the statement, measured by the homology of the chain complex
serves as a nice quantification of the complicatedness of the underlying region . These are the beginnings and main motivations of de Rham cohomology.
Decomposition theorem
It can be shown that any stationary flux that is twice continuously differentiable in and vanishes sufficiently fast for can be decomposed uniquely into an irrotational part and a source-free part . Moreover, these parts are explicitly determined by the respective source densities (see above) and circulation densities (see the article Curl):
For the irrotational part one has
with
The source-free part, , can be similarly written: one only has to replace the scalar potential by a vector potential and the terms by , and the source density by the circulation density .
This "decomposition theorem" is a by-product of the stationary case of electrodynamics. It is a special case of the more general Helmholtz decomposition, which works in dimensions greater than three as well.
In arbitrary finite dimensions
The divergence of a vector field can be defined in any finite number of dimensions. If
in a Euclidean coordinate system with coordinates , define
In the 1D case, reduces to a regular function, and the divergence reduces to the derivative.
For any , the divergence is a linear operator, and it satisfies the "product rule"
for any scalar-valued function .
Relation to the exterior derivative