\[ \DeclareMathOperator{\div}{div} \DeclareMathOperator{\grad}{grad} \DeclareMathOperator{\tr}{tr} \]

Let \(X\) be a smooth vector field on a Riemannian manifold \((M,g)\). The divergence is defined by

\begin{equation*} \div X := *^{-1}d(X\lrcorner dV_g), \end{equation*}

where \(d\) denotes the exterior derivative , \(dV_g\) the Riemannian volume form , \(\lrcorner\) the interior multiplication and \(*\) is the Hodge star operator .

Remarks
  • The term divergence is used because of it geometric interpretation. It describes the measure of the tendency of a vector field to “diverge”.
  • Equivalently the divergence suffices \begin{equation*} d(X\lrcorner dV_g)=(\div X)dV_g. \end{equation*}
  • The divergence may be defined with a Levi-Civita connection.

Properties Link to heading

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