Editing Divergence (section)

=== General coordinates ===
Using [[Einstein notation]] we can consider the divergence in [[Curvilinear coordinates|general coordinates]], which we write as {{math|''x''<sup>1</sup>, …, ''x''<sup>''i''</sup>, …, ''x''<sup>''n''</sup>}}, where {{mvar|n}} is the number of dimensions of the domain. Here, the upper index refers to the number of the coordinate or component, so {{math|''x''<sup>2</sup>}} refers to the second component, and not the quantity {{mvar|x}} squared. The index variable {{mvar|i}} is used to refer to an arbitrary component, such as {{math|''x''<sup>''i''</sup>}}. The divergence can then be written via the [https://www.genealogy.math.ndsu.nodak.edu/id.php?id=59087 Voss]-[[Hermann Weyl|Weyl]] formula,<ref>{{cite web|last1=Grinfeld|first1=Pavel|title=The Voss-Weyl Formula (Youtube link)|website=[[YouTube]] |date=16 April 2014 |url=https://www.youtube.com/watch?v=BD2AiFk651E&list=PLlXfTHzgMRULkodlIEqfgTS-H1AY_bNtq&index=23| archive-url=https://ghostarchive.org/varchive/youtube/20211211/BD2AiFk651E| archive-date=2021-12-11 | url-status=live|access-date=9 January 2018|language=en}}{{cbignore}}</ref> as:

<math display="block">\operatorname{div}(\mathbf{F}) = \frac{1}{\rho} \frac{\partial {\left(\rho \, F^i\right)}}{\partial x^i},</math>

where <math>\rho</math> is the local coefficient of the [[volume element]] and {{math|''F<sup>i</sup>''}} are the components of <math>\mathbf{F} = F^i\mathbf{e}_i</math> with respect to the local '''unnormalized''' [[Curvilinear coordinates#Covariant and contravariant bases|covariant basis]] (sometimes written as {{nowrap|<math>\mathbf{e}_i = \partial\mathbf{x} / \partial x^i</math>).}} The Einstein notation implies summation over {{mvar|i}}, since it appears as both an upper and lower index.

The volume coefficient {{mvar|ρ}} is a function of position which depends on the coordinate system. In Cartesian, cylindrical and spherical coordinates, using the same conventions as before, we have {{math|1=''ρ'' = 1}}, {{math|1=''ρ'' = ''r''}} and {{math|1=''ρ'' = ''r''<sup>2</sup> sin ''θ''}}, respectively. The volume can also be expressed as <math display="inline">\rho = \sqrt{\left|\det g_{ab}\right|}</math>, where {{math|''g<sub>ab</sub>''}} is the [[metric tensor]]. The [[determinant]] appears because it provides the appropriate invariant definition of the volume, given a set of vectors. Since the determinant is a scalar quantity which doesn't depend on the indices, these can be suppressed, writing {{nowrap|<math display="inline">\rho = \sqrt{\left|\det g\right|}</math>.}} The absolute value is taken in order to handle the general case where the determinant might be negative, such as in pseudo-Riemannian spaces. The reason for the square-root is a bit subtle: it effectively avoids double-counting as one goes from curved to Cartesian coordinates, and back. The volume (the determinant) can also be understood as the [[Jacobian matrix and determinant|Jacobian]] of the transformation from Cartesian to curvilinear coordinates, which for {{math|1=''n'' = 3}} gives {{nowrap|<math display="inline">\rho = \left| \frac{\partial(x,y,z)}{\partial (x^1,x^2,x^3)}\right|</math>.}}

Some conventions expect all local basis elements to be normalized to unit length, as was done in the previous sections. If we write <math>\hat{\mathbf{e}}_i</math> for the normalized basis, and <math>\hat{F}^i</math> for the components of {{math|'''F'''}} with respect to it, we have that 
<math display="block">\mathbf{F} = F^i \mathbf{e}_i =
F^i \|{\mathbf{e}_i }\| \frac{\mathbf{e}_i}{\| \mathbf{e}_i \|} =
F^i \sqrt{g_{ii}} \, \hat{\mathbf{e}}_i =
\hat{F}^i \hat{\mathbf{e}}_i,</math>
using one of the properties of the metric tensor. By dotting both sides of the last equality with the contravariant element {{nowrap|<math>\hat{\mathbf{e}}^i</math>,}} we can conclude that <math display="inline">F^i = \hat{F}^i / \sqrt{g_{ii}}</math>. After substituting, the formula becomes:

<math display="block">\operatorname{div}(\mathbf{F}) = \frac 1{\rho} \frac{\partial \left(\frac{\rho}{\sqrt{g_{ii}}}\hat{F}^i\right)}{\partial x^i} =
 \frac 1{\sqrt{\det g}} \frac{\partial \left(\sqrt{\frac{\det g}{g_{ii}}}\,\hat{F}^i\right)}{\partial x^i}.</math>

See ''{{section link||In curvilinear coordinates}}'' for further discussion.