Editing Divergence (section)

== Properties ==
{{Main|Vector calculus identities}}

The following properties can all be derived from the ordinary differentiation rules of [[calculus]]. Most importantly, the divergence is a [[linear operator]], i.e.,

:<math>\operatorname{div}(a\mathbf{F} + b\mathbf{G}) = a \operatorname{div} \mathbf{F} + b \operatorname{div} \mathbf{G}</math>

for all vector fields {{math|'''F'''}} and {{math|'''G'''}} and all [[real number]]s {{math|''a''}} and {{math|''b''}}.

There is a [[product rule]] of the following type: if {{mvar|φ}} is a scalar-valued function and {{math|'''F'''}} is a vector field, then

:<math>\operatorname{div}(\varphi \mathbf{F}) = \operatorname{grad} \varphi \cdot \mathbf{F} + \varphi \operatorname{div} \mathbf{F},</math>

or in more suggestive notation

:<math>\nabla\cdot(\varphi \mathbf{F}) = (\nabla\varphi) \cdot \mathbf{F} + \varphi (\nabla\cdot\mathbf{F}).</math>

Another product rule for the [[cross product]] of two vector fields {{math|'''F'''}} and {{math|'''G'''}} in three dimensions involves the [[Curl (mathematics)|curl]] and reads as follows:

:<math>\operatorname{div}(\mathbf{F}\times\mathbf{G}) = \operatorname{curl} \mathbf{F} \cdot\mathbf{G} - \mathbf{F} \cdot \operatorname{curl} \mathbf{G},</math>

or

:<math>\nabla\cdot(\mathbf{F}\times\mathbf{G}) = (\nabla\times\mathbf{F})\cdot\mathbf{G} - \mathbf{F}\cdot(\nabla\times\mathbf{G}).</math>

The [[Laplacian]] of a [[scalar field]] is the divergence of the field's [[gradient]]:
:<math>\operatorname{div}(\operatorname{grad}\varphi) = \Delta\varphi.</math>

The divergence of the [[Curl (mathematics)|curl]] of any vector field (in three dimensions) is equal to zero: 
:<math>\nabla\cdot(\nabla\times\mathbf{F})=0.</math>

If a vector field {{math|'''F'''}} with zero divergence is defined on a ball in {{math|'''R'''<sup>3</sup>}}, then there exists some vector field {{math|'''G'''}} on the ball with {{math|'''F''' {{=}} curl '''G'''}}. For regions in {{math|'''R'''<sup>3</sup>}} 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 (mathematics)|homology]] of the [[chain complex]]

:<math>\{ \text{scalar fields on } U \} ~ \overset{\operatorname{grad}}{\rarr} ~ \{ \text{vector fields on } U \} ~ \overset{\operatorname{curl}}{\rarr} ~ \{ \text{vector fields on } U \} ~ \overset{\operatorname{div}}{\rarr} ~ \{ \text{scalar fields on } U \}</math>

serves as a nice quantification of the complicatedness of the underlying region {{math|''U''}}. These are the beginnings and main motivations of [[de Rham cohomology]].