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In Vector Calculus , the divergence is an operator that measures a Vector Field 's tendency to originate from or converge upon a given point. For instance, for a Vector Field that denotes the Velocity of air expanding as it is heated, the divergence of the velocity field would have a positive value because the air is expanding. Conversely, if the air is cooling and contracting, the divergence would be negative.

A vector field which has zero divergence everywhere is called Solenoidal .


DEFINITION


Let ''x, y, z'' be a system of Cartesian Coordinates on a 3-dimensional Euclidean Space , and let i, '''j''', '''k''' be the corresponding Basis of Unit Vector s.

The divergence of a Continuous ly Differentiable Vector Field F = ''F1'' '''i''' + ''F2'' '''j''' + ''F3'' '''k''' is defined to be the Scalar -valued function:

:\operatorname{div}\,\mathbf{F} =
abla\cdot\mathbf{F}
= rac{\partial F_1}{\partial x}
+ rac{\partial F_2}{\partial y}
+ rac{\partial F_3}{\partial z}.

Although expressed in terms of coordinates, the result is invariant under Orthogonal Transformation s, as the physical interpretation suggests.

The common notation for the divergence ∇·'''F''' is a convenient mnemonic, where the dot denotes something just reminiscent of the ), apply them to the components of '''F''', and sum the results.


PHYSICAL INTERPRETATION


In physical terms, the divergence of a vector field is the extent to which the vector field flow behaves like a source or a sink at a given point. Indeed, an alternative, but logically equivalent definition, gives the divergence as the Derivative of the Net Flow of the vector field across the surface of a small Sphere relative to the Volume of the sphere. Formally,

:( \operatorname{div}\,\mathbf{F}) (p) =
\lim_{r ightarrow 0}
\int_{S(r)} {\mathbf{F}\cdot\mathbf{n}dS \over rac{4}{3} \pi r^3 }

where ''S''(''r'') denotes the sphere of radius ''r'' about a point ''p'' in R3, and the integral is a Surface Integral taken with respect to '''n''', the normal to that sphere.

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

The intuition that the sum of all sources minus the sum of all sinks should give the net flow outwards of a region is made precise by the Divergence Theorem .


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.

:\operatorname{div}( a\mathbf{F} + b\mathbf{G} )
= a\;\operatorname{div}( \mathbf{F} )
+ b\;\operatorname{div}( \mathbf{G} )

for all vector fields F and '''G''' and all Real Number s ''a'' and ''b''.

There is a Product Rule of the following type: if φ is a scalar valued function and F is a vector field, then

:\operatorname{div}( arphi \mathbf{F})
= \operatorname{grad}( arphi) \cdot \mathbf{F}
+ arphi \;\operatorname{div}(\mathbf{F}),

or in more suggestive notation

:
abla\cdot( arphi \mathbf{F})
= (
abla arphi) \cdot \mathbf{F}
+ arphi \;(
abla\cdot\mathbf{F}).

Another product rule for the Cross Product of two vector fields F and '''G''' in three dimensions involves the Curl and reads as follows:

:\operatorname{div}(\mathbf{F} imes\mathbf{G})
= \operatorname{curl}(\mathbf{F})\cdot\mathbf{G}
\;-\; \mathbf{F} \cdot \operatorname{curl}(\mathbf{G}),

or

:
abla\cdot(\mathbf{F} imes\mathbf{G})
= (
abla imes\mathbf{F})\cdot\mathbf{G}
- \mathbf{F}\cdot(
abla imes\mathbf{G}).

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 constant and equal to zero. Conversely, if you have a vector field F with zero divergence defined on a ball in '''R'''3, say, then there exists some vector field '''G''' on the ball with F = curl('''G'''). For regions in '''R'''3 more complicated than balls, this latter statement might not be true anymore (see Poincaré Lemma ). Indeed, the degree of ''failure'' of the truth of the statement, measured by the Homology of the Chain Complex

: \{\mbox{scalar fields on }U\} \;
:: o\{\mbox{vector fields on }U\} \;
::: o\{\mbox{vector fields on }U\} \;
:::: o\{\mbox{scalar fields on }U\} \;

(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 .


RELATION WITH THE EXTERIOR DERIVATIVE

One can establish a parallel between the divergence and a particular case of the exterior derivative, when it takes a 2-form to a 3-form in R3.
If we define:
:\alpha=F_1\ dy\wedge dz + F_2\ dz\wedge dx + F_3\ dx\wedge dy
its Exterior Derivative d\alpha is given by
:d\alpha = \left( rac{\partial F_1}{\partial x}
+ rac{\partial F_2}{\partial y}
+ rac{\partial F_3}{\partial z} ight) dx\wedge dy\wedge dz


GENERALIZATIONS


The divergence of a vector field can be defined in any number of dimensions. If
:\mathbf{F}=(F_1, F_2, \dots, F_n),

define

:\operatorname{div}\,\mathbf{F} =
abla\cdot\mathbf{F}
= rac{\partial F_1}{\partial x_1}
+ rac{\partial F_2}{\partial x_2}+\cdots
+ rac{\partial F_n}{\partial x_n}.

For any ''n'', the divergence is a linear operator, and it satisfies the "product rule"

:
abla\cdot( arphi \mathbf{F})
= (
abla arphi) \cdot \mathbf{F}
+ arphi \;(
abla\cdot\mathbf{F}).

for any scalar-valued function φ.


SEE ALSO