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''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" />''
+
''of a vector field $ \mathbf{a} $ at a point $ x = (x^{1},\ldots,x^{n}) $''
  
 
The scalar field
 
The scalar field
 +
$$
 +
x \mapsto \sum_{i = 1}^{n} \frac{\partial}{\partial x^{i}} [{a^{i}}(x)],
 +
$$
 +
where the $ a^{i} $’s are the components of the vector field $ \mathbf{a} $.
  
<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/d0336003.png" /></td> </tr></table>
+
The divergence of a vector field $ \mathbf{a} $ at a point $ x $ is denoted by $ (\operatorname{div} \mathbf{a})(x) $ or by the inner product $ \langle \nabla,\mathbf{a} \rangle (x) $ of the [[Hamilton operator|Hamilton operator]] $ \nabla \stackrel{\text{df}}{=} \left( \dfrac{\partial}{\partial x^{1}},\ldots,\dfrac{\partial}{\partial x^{n}} \right) $ and the vector $ \mathbf{a}(x) $.
  
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" />.
+
If the vector field $ \mathbf{a} $ is the field of velocities of a stationary flow of a non-compressible liquid, then $ (\operatorname{div} \mathbf{a})(x) $ coincides with the intensity of the source (when $ (\operatorname{div} \mathbf{a})(x) > 0 $) or the sink (when $ (\operatorname{div} \mathbf{a})(x) < 0 $) at the point $ x $.
 
 
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
 
The integral
 +
$$
 +
\int_{E} \operatorname{div}(\rho ~ \mathbf{a}) ~ \mathrm{d}{x},
 +
$$
 +
where $ \rho $ is the density of the liquid computed for the $ n $-dimensional domain $ E $, is equal to the amount of the liquid ‘issuing’ from $ E $ per unit time. This amount (cf. [[Ostrogradski formula|Ostrogradski’s Formula]]) coincides with the magnitude
 +
$$
 +
\int_{\partial E} \langle \mathbf{N},\rho ~ \mathbf{a} \rangle ~ \mathrm{d}{S} = \sum_{i = 1}^{n} \int_{\partial E} N_{i} \rho a^{i} ~ \mathrm{d}{S},
 +
$$
 +
where $ \mathbf{N} = (N_{1},\ldots,N_{n}) $ denotes the exterior unit normal vector to $ \partial E $, and $ \mathrm{d}{S} $ is the area element of $ \partial E $. The divergence $ (\operatorname{div} \mathbf{a})(x) $ is then the derivative with respect to the rate of the flow $ \mathbf{a} $ across the closed boundary surface $ \partial E $:
 +
$$
 +
(\operatorname{div} \mathbf{a})(x) = \lim_{E \to \{ x \}} \frac{1}{\operatorname{Vol}(E)} \int_{\partial E} \langle \mathbf{N},\mathbf{a} \rangle ~ \mathrm{d}{S}
 +
$$
 +
Thus, the divergence is invariant with respect to the choice of coordinate system.
  
<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/d03360015.png" /></td> </tr></table>
+
In curvilinear coordinates $ y = (y^{1},\ldots,y^{n}) $, we have
 
+
$$
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
+
(\operatorname{div} \mathbf{a})(y) = \frac{1}{\sqrt{g}} \sum_{i = 1}^{n} \frac{\partial}{\partial y^{i}} \left[ \sqrt{g} a^{i} \right], \quad \text{with} \quad
 
+
g \stackrel{\text{df}}{=} \det([g_{ij}]) \quad \text{and} \quad g_{ij} \stackrel{\text{df}}{=} \sum_{\alpha = 1}^{n} \frac{\partial y^{\alpha}}{\partial x^{i}} \frac{\partial y^{\alpha}}{\partial x^{j}}, \qquad (\star)
<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/d03360020.png" /></td> </tr></table>
+
$$
 
 
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:
 
 
 
<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/d03360027.png" /></td> </tr></table>
 
 
 
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>
 
 
 
<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/d03360032.png" /></td> </tr></table>
 
 
 
 
where
 
where
 
+
$ \displaystyle \mathbf{a}(y) \stackrel{\text{df}}{=} \sum_{i = 1}^{n} {a^{i}}(y) ~ {\mathbf{s}_{i}}(y) $, and $ {\mathbf{s}_{i}}(y) $ is the unit tangent vector to the $ i $-th coordinate line at the point $ y $:
<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/d03360033.png" /></td> </tr></table>
+
$$
 
+
{\mathbf{s}_{i}}(y) \stackrel{\text{df}}{=} \frac{1}{\sqrt{g_{ii}}} \frac{\partial y}{\partial x^{i}}.
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" />:
+
$$
 
 
<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/d03360037.png" /></td> </tr></table>
 
 
 
 
The divergence of a tensor field
 
The divergence of a tensor field
 
+
$$
<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/d03360038.png" /></td> </tr></table>
+
x \mapsto a(x) = \left\{ {a^{i_{1} \ldots i_{p}}_{j_{1} \ldots j_{q}}}(x) ~ \middle| ~ i_{\alpha},j_{\beta} \in \mathbb{N}_{\leq n} \right\}
 
+
$$
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
+
of type $ (p,q) $, defined on an $ n $-dimensional manifold with an affine connection, is defined with the aid of the corresponding absolute (covariant) derivatives of the components of $ a(x) $, with subsequent convolution (contraction), and is a tensor of type $ (p - 1,q) $ 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>
+
{b^{i_{1} \ldots i_{s - 1} i_{s + 1} \ldots i_{p}}_{j_{1} \ldots j_{q}}}(x) = \sum_{k = 1}^{n} {\nabla_{k} a^{i_{1} \ldots i_{s - 1} k i_{s + 1} \ldots i_{p}}_{j_{1} \ldots j_{q}}}(x), \qquad k,i_{\alpha},j_{\beta} \in \mathbb{N}_{\leq n}.
 
+
$$
<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/d03360044.png" /></td> </tr></table>
+
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'''.
 
 
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====
 
====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>
 
 
  
 +
<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====
 
====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]].
+
The Hamilton operator is usually called the '''nabla operator''', after the symbol for it, $ \nabla $. Ostrogradski’s formula is better known as the '''Gauss–Ostrogradski formula''' or the '''Gauss formula'''.
  
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
+
For other vector differentiation operators, see [[Curl|Curl]]; [[Gradient|Gradient]]. For relations between these, see also [[Vector analysis|Vector analysis]].
  
<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>
+
Let $ M $ be an $ n $-dimensional manifold and $ \omega $ a volume element on $ M $. The Lie derivative $ {L_{X}}(\omega) $ is then also a differential $ n $-form, and so $ {L_{X}}(\omega) = f_{X} \omega $ for some function $ f_{X} $ on $ M $. This function is the divergence $ \operatorname{div}(X) $ of $ X $ with respect to the volume element $ \omega $. If $ g $ is a Riemannian metric on $ M $, then the divergence of $ X $ as defined by $ (\star) $ above is the divergence of $ X $ with respect to the volume element $ \omega_{g} \stackrel{\text{df}}{=} \sqrt{\det(g)} \cdot \mathrm{d}{x^{1}} \wedge \cdots \wedge \mathrm{d}{x^{n}} $ defined by $ g $. For any function $ f $, note that $ f \omega $ is an $ n $-form, so $ \displaystyle \int_{M} f \omega $ is defined — this is just the integral of $ f $ with respect to the volume element $ \omega $. If $ M $ is compact, then Green’s theorem says that
 +
$$
 +
\int_{M} \operatorname{div}(X) ~ \omega = 0.
 +
$$
 +
Still another notation for the divergence of an $ n $-tuple $ \mathbf{a} = (a^{1},\ldots,a^{n}) $ of functions of $ x_{1},\ldots,x_{n} $ (or of a vector field) is $ \nabla \cdot \mathbf{a} $.
  
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====
  
====References====
+
<table>
<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>
+
<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>

Latest revision as of 05:04, 11 December 2016

of a vector field $ \mathbf{a} $ at a point $ x = (x^{1},\ldots,x^{n}) $

The scalar field $$ x \mapsto \sum_{i = 1}^{n} \frac{\partial}{\partial x^{i}} [{a^{i}}(x)], $$ where the $ a^{i} $’s are the components of the vector field $ \mathbf{a} $.

The divergence of a vector field $ \mathbf{a} $ at a point $ x $ is denoted by $ (\operatorname{div} \mathbf{a})(x) $ or by the inner product $ \langle \nabla,\mathbf{a} \rangle (x) $ of the Hamilton operator $ \nabla \stackrel{\text{df}}{=} \left( \dfrac{\partial}{\partial x^{1}},\ldots,\dfrac{\partial}{\partial x^{n}} \right) $ and the vector $ \mathbf{a}(x) $.

If the vector field $ \mathbf{a} $ is the field of velocities of a stationary flow of a non-compressible liquid, then $ (\operatorname{div} \mathbf{a})(x) $ coincides with the intensity of the source (when $ (\operatorname{div} \mathbf{a})(x) > 0 $) or the sink (when $ (\operatorname{div} \mathbf{a})(x) < 0 $) at the point $ x $.

The integral $$ \int_{E} \operatorname{div}(\rho ~ \mathbf{a}) ~ \mathrm{d}{x}, $$ where $ \rho $ is the density of the liquid computed for the $ n $-dimensional domain $ E $, is equal to the amount of the liquid ‘issuing’ from $ E $ per unit time. This amount (cf. Ostrogradski’s Formula) coincides with the magnitude $$ \int_{\partial E} \langle \mathbf{N},\rho ~ \mathbf{a} \rangle ~ \mathrm{d}{S} = \sum_{i = 1}^{n} \int_{\partial E} N_{i} \rho a^{i} ~ \mathrm{d}{S}, $$ where $ \mathbf{N} = (N_{1},\ldots,N_{n}) $ denotes the exterior unit normal vector to $ \partial E $, and $ \mathrm{d}{S} $ is the area element of $ \partial E $. The divergence $ (\operatorname{div} \mathbf{a})(x) $ is then the derivative with respect to the rate of the flow $ \mathbf{a} $ across the closed boundary surface $ \partial E $: $$ (\operatorname{div} \mathbf{a})(x) = \lim_{E \to \{ x \}} \frac{1}{\operatorname{Vol}(E)} \int_{\partial E} \langle \mathbf{N},\mathbf{a} \rangle ~ \mathrm{d}{S} $$ Thus, the divergence is invariant with respect to the choice of coordinate system.

In curvilinear coordinates $ y = (y^{1},\ldots,y^{n}) $, we have $$ (\operatorname{div} \mathbf{a})(y) = \frac{1}{\sqrt{g}} \sum_{i = 1}^{n} \frac{\partial}{\partial y^{i}} \left[ \sqrt{g} a^{i} \right], \quad \text{with} \quad g \stackrel{\text{df}}{=} \det([g_{ij}]) \quad \text{and} \quad g_{ij} \stackrel{\text{df}}{=} \sum_{\alpha = 1}^{n} \frac{\partial y^{\alpha}}{\partial x^{i}} \frac{\partial y^{\alpha}}{\partial x^{j}}, \qquad (\star) $$ where $ \displaystyle \mathbf{a}(y) \stackrel{\text{df}}{=} \sum_{i = 1}^{n} {a^{i}}(y) ~ {\mathbf{s}_{i}}(y) $, and $ {\mathbf{s}_{i}}(y) $ is the unit tangent vector to the $ i $-th coordinate line at the point $ y $: $$ {\mathbf{s}_{i}}(y) \stackrel{\text{df}}{=} \frac{1}{\sqrt{g_{ii}}} \frac{\partial y}{\partial x^{i}}. $$ The divergence of a tensor field $$ x \mapsto a(x) = \left\{ {a^{i_{1} \ldots i_{p}}_{j_{1} \ldots j_{q}}}(x) ~ \middle| ~ i_{\alpha},j_{\beta} \in \mathbb{N}_{\leq n} \right\} $$ of type $ (p,q) $, defined on an $ n $-dimensional manifold with an affine connection, is defined with the aid of the corresponding absolute (covariant) derivatives of the components of $ a(x) $, with subsequent convolution (contraction), and is a tensor of type $ (p - 1,q) $ with components $$ {b^{i_{1} \ldots i_{s - 1} i_{s + 1} \ldots i_{p}}_{j_{1} \ldots j_{q}}}(x) = \sum_{k = 1}^{n} {\nabla_{k} a^{i_{1} \ldots i_{s - 1} k i_{s + 1} \ldots i_{p}}_{j_{1} \ldots j_{q}}}(x), \qquad k,i_{\alpha},j_{\beta} \in \mathbb{N}_{\leq n}. $$ 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

[1] N.E. Kochin, “Vector calculus and fundamentals of tensor calculus”, Moscow (1965). (In Russian)
[2] P.K. [P.K. Rashevskii] Rashewski, “Riemannsche Geometrie und Tensoranalyse”, Deutsch. Verlag Wissenschaft. (1959). (Translated from Russian)

Comments

The Hamilton operator is usually called the nabla operator, after the symbol for it, $ \nabla $. Ostrogradski’s formula is better known as the Gauss–Ostrogradski formula or the Gauss formula.

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

Let $ M $ be an $ n $-dimensional manifold and $ \omega $ a volume element on $ M $. The Lie derivative $ {L_{X}}(\omega) $ is then also a differential $ n $-form, and so $ {L_{X}}(\omega) = f_{X} \omega $ for some function $ f_{X} $ on $ M $. This function is the divergence $ \operatorname{div}(X) $ of $ X $ with respect to the volume element $ \omega $. If $ g $ is a Riemannian metric on $ M $, then the divergence of $ X $ as defined by $ (\star) $ above is the divergence of $ X $ with respect to the volume element $ \omega_{g} \stackrel{\text{df}}{=} \sqrt{\det(g)} \cdot \mathrm{d}{x^{1}} \wedge \cdots \wedge \mathrm{d}{x^{n}} $ defined by $ g $. For any function $ f $, note that $ f \omega $ is an $ n $-form, so $ \displaystyle \int_{M} f \omega $ is defined — this is just the integral of $ f $ with respect to the volume element $ \omega $. If $ M $ is compact, then Green’s theorem says that $$ \int_{M} \operatorname{div}(X) ~ \omega = 0. $$ Still another notation for the divergence of an $ n $-tuple $ \mathbf{a} = (a^{1},\ldots,a^{n}) $ of functions of $ x_{1},\ldots,x_{n} $ (or of a vector field) is $ \nabla \cdot \mathbf{a} $.

References

[a1] D.E. Bourne, P.C. Kendall, “Vector analysis and Cartesian tensors”, Nelson & Sons, Sunbury-on-Thames (1977). Zbl 0781.53013
How to Cite This Entry:
Divergence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Divergence&oldid=39958
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article