Let \((M,g)\) be a smooth compact Riemannian manifold with boundary, \(F\) a covariant \(k\)-tensor and \(G\) a covariant \((k+1)\)-tensor. Then
\begin{equation*} \int_{M} \langle \nabla F, G\rangle dV_g = \int_{\partial M} \langle F\otimes N^\flat, G\rangle dV_{\hat{g}}- \int_{M} \langle F, \div G\rangle dV_g, \end{equation*}where \(\langle \cdot, \cdot\rangle\) denotes the inner product of tensors , \(\nabla F\) the total covariant derivative of \(F\), \(N\) is the outward-pointing unit normal vector along \(\partial M\), \(\hat{g}\) the induced Riemannian metric on \(\partial M\) and \(\div G\) the tensor divergence . [1, Problem 5-16]
Remarks
In local coordinates the above equality may be written as
\begin{align} \int_{M} F_{i_1\ldots i_k;j}&G^{i_1\ldots i_kj}dV_g \\ &=\int_{\partial M} F_{i_1\ldots i_k}G^{i_1\ldots i_kj}N_j dV_{\hat{g}}-\int_{M} F_{i_1\ldots i_k}{G^{i_1\ldots i_kj}}_{;j}. \end{align}References Link to heading
- J. Lee, Introduction to Smooth Manifolds. New York ; London: Springer, 2013.