= Divergence =

'''Divergence''' measures the distortion of a [[Calculus/VectorField|vector field]].

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== Description ==

The divergence of a [[Calculus/VectorField|vector field]] given as ''F = <P(x,y), Q(x,y)>'' is calculated as:

{{attachment:div1.svg}}

The divergence of ''F'' given as ''<P(x,y,z), Q(x,y,z), R(x,y,z)>'' is calculated as:

{{attachment:div2.svg}}

Note that divergence returns a scalar value.



=== Properties ===

The divergence of the [[Calculus/Curl|curl]] of a vector field is always 0.

A vector field is [[Calculus/VectorField#Source-Free_Fields|source-free]] if there is zero divergence.

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== Conservative Fields ==

A [[Calculus/VectorField|vector field]] is [[Calculus/VectorField#Cross-Partial_Property_of_Conservative_Fields|conservative]] if it is path independent. There is necessarily at least one function ''f'' satisfying ''F = ∇f''. In this case, divergence can be calculated using the [[Calculus/VectorField#Laplace_Operator|Laplace operator]].

For a vector field given as ''F = <P(x,y), Q(x,y)>'', applying the Laplace operator looks like:

{{attachment:laplace1.svg}}

where ''f,,xx,, = ∂^2^f/∂x^2^'' and ''f,,yy,, = ∂^2^f/∂y^2^''.

For a vector field given as ''F=<P,Q,R>'', applying the Laplace operator looks like:

{{attachment:laplace2.svg}}

where ''f,,zz,, = ∂^2^f/∂z^2^''.



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CategoryRicottone