Coordinate and tensor divergence theorems

The expression previously obtained for the divergence theorem was

\(\displaystyle \begin{aligned}\int_{V}\mathrm{div}(u)\mathrm{d}V & =\int_{\partial V}i_{u}\mathrm{d}V\\ & =\int_{\partial V}\left\langle u,\hat{n}\right\rangle \mathrm{d}S, \end{aligned} \)

where \({V}\) is an \({n}\)-dimensional compact submanifold of \({M^{n}}\), \({\hat{n}}\) is the unit normal vector to \({\partial V}\), and \({\mathrm{d}S\equiv i_{\hat{n}}\mathrm{d}V}\) is the induced volume element (“surface element”) for \({\partial V}\). If we choose an orthonormal frame with \({e_{1}=\hat{n}}\) on \({\partial V}\), the divergence theorem can be written

\begin{aligned}\int_{V}\mathrm{div}(u)\mathrm{d}V & =\int_{\partial V}u^{1}\mathrm{d}S,\end{aligned}

and if we can choose coordinates with \({x^{1}}\) constant on \({\partial V}\) and normal to it, the divergence theorem can be written

\begin{aligned}\int_{V}\partial_{\lambda}\left(\sqrt{g}u^{\lambda}\right)\mathrm{d}^{n}x & =\int_{\partial V}\sqrt{g}\mathrm{d}x^{1}\left(u\right)\mathrm{d}^{n-1}x\\
& =\int_{\partial V}u^{1}\sqrt{g}\mathrm{d}^{n-1}x,
\end{aligned}

where \({\mathrm{d}^{n}x\equiv\mathrm{d}x^{1}\wedge\cdots\wedge\mathrm{d}x^{n}}\) and \({\mathrm{d}^{n-1}x\equiv\mathrm{d}x^{2}\wedge\cdots\wedge\mathrm{d}x^{n}}\).

Since the “divergence” of a tensor \({T}\) with order greater than 1 is tensor-valued, and the parallel transport of tensors is path-dependent, we cannot in general integrate to get a divergence theorem for tensors. In the case of a flat metric and zero torsion however, we can choose coordinates whose coordinate frame is orthonormal, so that the frame is its own parallel transport, i.e. \({\nabla_{v}\left(\beta^{\mu}\right)=0}\). For e.g. a tensor \({T^{ab}}\), we can then define a coordinate-dependent vector \({J^{\mu}}\) for each index \({\mu}\)

\begin{aligned}\require{cancel}J^{\mu} & \equiv T\left(\beta^{\mu},\quad\right)\\
\Rightarrow\left(J^{\mu}\right)^{b} & =T^{\mu b}\\
\Rightarrow\nabla_{v}J^{\mu} & \overset{\cancel{RT}}{=}\beta^{\mu}\nabla_{v}T\\
\Rightarrow\nabla_{b}\left(J^{\mu}\right)^{b} & \overset{\cancel{RT}}{=}\nabla_{b}T^{\mu b}\\
\Rightarrow\int_{V}\nabla_{b}T^{\mu b}\mathrm{d}V & \overset{\cancel{RT}}{=}\int_{V}\nabla_{b}\left(J^{\mu}\right)^{b}\mathrm{d}V\\
& =\int_{\partial V}T^{\mu}{}_{b}\hat{n}^{b}\mathrm{d}S.
\end{aligned}

For arbitrary coordinates, the components of the coordinate frame are by definition constant, i.e. \({\partial_{v}\left(\mathrm{d}x^{\mu}\right)=0}\); we can therefore write

\begin{aligned}\require{cancel}\sqrt{g}J^{\mu} & \equiv\sqrt{g}T\left(\mathrm{d}x^{\mu},\quad\right)\\
\Rightarrow\partial_{\nu}\left(\sqrt{g}J^{\mu}\right)^{\nu} & =\partial_{\nu}\left(\sqrt{g}T^{\mu\nu}\right)\\
\Rightarrow\int_{V}\partial_{\nu}\left(\sqrt{g}T^{\mu\nu}\right)\mathrm{d}^{n}x & =\int_{V}\partial_{\nu}\left(\sqrt{g}J^{\mu}\right)^{\nu}\mathrm{d}^{n}x\\
& =\int_{V}\nabla_{b}\left(J^{\mu}\right)^{b}\mathrm{d}V\\
& =\int_{\partial V}T^{\mu}{}_{b}\hat{n}^{b}\mathrm{d}S.
\end{aligned}

This relation remains true in the presence of both curvature and torsion, however it is important to note that \({\partial_{\nu}\left(\sqrt{g}T^{\mu\nu}\right)}\) is not a “divergence” and \({T^{\mu b}=\left(J^{\mu}\right)^{b}}\) is coordinate-dependent. In the special case of an anti-symmetric tensor under zero torsion, we can write

\begin{aligned}\require{cancel}\int_{V}\nabla_{\nu}F^{\mu\nu}\mathrm{d}V & \overset{\cancel{T}}{=}\int_{V}\partial_{\nu}\left(\sqrt{g}F^{\mu\nu}\right)\mathrm{d}^{n}x\\
& =\int_{\partial V}F^{\mu}{}_{b}\hat{n}^{b}\mathrm{d}S.
\end{aligned}

An Illustrated Handbook