View source for Divergence

''of a vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d0336001.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d0336002.png" />''

The scalar field

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d0336004.png" /> are the components of the vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d0336005.png" />.

The divergence is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d0336006.png" /> or by the inner product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d0336007.png" /> of the [[Hamilton operator|Hamilton operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d0336008.png" /> and the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d0336009.png" />.

If the vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360010.png" /> is the field of velocities of a stationary flow of a non-compressible liquid, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360011.png" /> coincides with the intensity of the source (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360012.png" />) or the sink (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360013.png" />) at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360014.png" />.

The integral

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360016.png" /> is the density of the liquid computed for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360017.png" />-dimensional domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360018.png" />, is equal to the amount of the liquid  "issuing"  from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360019.png" /> in unit time. This amount (cf. [[Ostrogradski formula|Ostrogradski formula]]) coincides with the magnitude

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360021.png" /> is the unit exterior normal vector to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360022.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360023.png" /> is the area element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360024.png" />. The divergence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360025.png" /> is the derivative with respect to the rate of the flow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360026.png" /> across the closed surface:

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Thus, the divergence is invariant with respect to the choice of the coordinate system.

In curvilinear coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360030.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360031.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

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where

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and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360034.png" /> is the unit tangent vector to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360035.png" />-th coordinate line at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360036.png" />:

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The divergence of a tensor field

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of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360039.png" /> defined in a domain of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360040.png" />-dimensional manifold with an affine connection, is defined with the aid of the corresponding absolute (covariant) derivatives of the components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360041.png" />, with subsequent convolution (contraction), and is a tensor of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360042.png" /> with components

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360043.png" /></td> </tr></table>

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In tensor analysis and differential geometry a differential operator operating on the space of differential forms and connected with the operator of exterior differentiation is also called a divergence.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.E. Kochin,   "Vector calculus and fundamentals of tensor calculus" , Moscow  (1965)  (In Russian)  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.K. [P.K. Rashevskii] Rashewski,   "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)  {{MR|}} {{ZBL|}} </TD></TR></table>



====Comments====
The Hamilton operator is usually called nabla operator, after the symbol for it, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360045.png" />. Ostrogradski's formula is better known as the Gauss–Ostrogradski or Gauss formula.

For other vector differentiation operators see [[Curl|Curl]]; [[Gradient|Gradient]]. For relations between these see also [[Vector analysis|Vector analysis]].

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360046.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360047.png" />-dimensional manifold and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360048.png" /> a volume element on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360049.png" />. The Lie derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360050.png" /> is then also a differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360051.png" />-form and so <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360052.png" /> for some function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360053.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360054.png" />. This function is the divergence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360055.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360056.png" /> with respect to the volume element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360057.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360058.png" /> is a Riemannian metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360059.png" />, then the divergence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360060.png" /> as defined by (*) above is the divergence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360061.png" /> with respect to the volume element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360062.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360063.png" />. For any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360064.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360065.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360066.png" />-form, so <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360067.png" /> is defined — the integral of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360068.png" /> with respect to a volume element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360069.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360070.png" /> is compact, then Green's theorem says that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360071.png" /></td> </tr></table>

Still another notation for the divergence of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360072.png" />-tuple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360073.png" /> of functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360074.png" /> (or of a vector field) is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033600/d03360075.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.E. Bourne,   P.C. Kendall,   "Vector analysis and Cartesian tensors" , Nelson &amp; Sons , Sunbury-on-Thames  (1977)  {{MR|}} {{ZBL|0781.53013}} </TD></TR></table>