Editing Divergence (section)

==In curvilinear coordinates==
The appropriate expression is more complicated in [[Curvilinear coordinates#Grad, curl, div, Laplacian|curvilinear coordinates]]. The divergence of a vector field extends naturally to any [[differentiable manifold]] of dimension {{math|''n''}} that has a [[volume form]] (or [[density on a manifold|density]]) {{mvar|μ}}, e.g. a [[Riemannian manifold|Riemannian]] or [[Lorentzian manifold]]. Generalising the construction of a [[two-form]] for a vector field on {{math|'''R'''<sup>3</sup>}}, on such a manifold a vector field {{math|''X''}} defines an {{math|(''n'' − 1)}}-form {{math|1=''j'' = ''i''<sub>''X''</sub>&thinsp;''μ''}} obtained by contracting {{math|''X''}} with {{mvar|μ}}. The divergence is then the function defined by

:<math>dj = (\operatorname{div} X) \mu .</math>

The divergence can be defined in terms of the [[Lie derivative]] as

:<math>{\mathcal L}_X \mu = (\operatorname{div} X) \mu .</math>

This means that the divergence measures the rate of expansion of a unit of volume (a [[volume element]]) as it flows with the vector field.

On a [[pseudo-Riemannian manifold]], the divergence with respect to the volume can be expressed in terms of the [[Levi-Civita connection]] {{math|∇}}:

:<math>\operatorname{div} X = \nabla \cdot X = {X^a}_{;a} ,</math>

where the second expression is the contraction of the vector field valued 1-form {{math|∇''X''}} with itself and the last expression is the traditional coordinate expression from [[Ricci calculus]].

An equivalent expression without using a connection is

:<math>\operatorname{div}(X) = \frac{1}{\sqrt{\left|\det g \right|}} \, \partial_a \left(\sqrt{\left|\det g \right|} \, X^a\right),</math>

where {{mvar|g}} is the [[metric tensor|metric]] and <math>\partial_a</math> denotes the partial derivative with respect to coordinate {{math|''x''{{i sup|''a''}}}}. The square-root of the (absolute value of the [[determinant]] of the) metric appears because the divergence must be written with the correct conception of the [[volume]].  In curvilinear coordinates, the basis vectors are no longer orthonormal; the determinant encodes the correct idea of volume in this case. It appears twice, here, once, so that the <math>X^a</math> can be transformed into "flat space" (where coordinates are actually orthonormal), and once again so that <math>\partial_a</math> is also transformed into "flat space", so that finally, the "ordinary" divergence can be written with the "ordinary" concept of volume in flat space (''i.e.'' unit volume, ''i.e.'' one, ''i.e.'' not written down). The square-root appears in the denominator, because the derivative transforms in the opposite way ([[Covariance and contravariance of vectors|contravariantly]]) to the vector (which is [[Covariance and contravariance of vectors|covariant]]). This idea of getting to a "flat coordinate system" where local computations can be done in a conventional way is called a [[vielbein]]. A different way to see this is to note that the divergence is the [[codifferential]] in disguise. That is, the divergence corresponds to the expression <math>\star d\star</math> with <math>d</math> the [[differential of a function|differential]] and <math>\star</math> the [[Hodge star]]. The Hodge star, by its construction, causes the [[volume form]] to appear in all of the right places.