Let $(M,g)$ be a Riemannian manifold, and $E \to M$ a vector bundle over $M$ with fibre metric $h$ and compatible connection $\nabla$. The Laplace operator $\Delta : \Omega^k(E) \to \Omega^k(E)$ is defined on $E$-valued forms by \[ \Delta = \mathrm{d}\mathrm{d}^* + \mathrm{d}^* \mathrm{d}. \]

Another Laplacian on forms is the rough Laplacian, $\nabla^* \nabla$. In an orthonormal frame $(e_i)_{i=1}^n$ of $TM$ with dual frame $(\omega^i)_{i=1}^n$, the rough Laplacian may be expressed in coordinates by \begin{align*} \nabla^* \nabla &= -e^*(\omega^i)\nabla_{e_i} e(\omega^j) \nabla_{e_j} \\ &= -e^*(\omega^i) e(\omega^j) \nabla_{e_i} \nabla_{e_j} + e(\nabla_{e_i} \omega^j) \nabla_{e_j} \\ &= -\nabla_{e_i} \nabla_{e_i} + (-e^*(\omega^i) e(\nabla_{e_i} e_j) \nabla_{e_j}). &= -\nabla_{e_i} \nabla_{e_i} - \langle \nabla_{e_i} e_j, e_i \rangle \nabla_{e_j}. \end{align*} To simplify further, remark that \begin{align*} \langle \nabla_{e_i} e_j, e_i \rangle_{L^2} \nabla_{e_j} &= -\langle e_j, \nabla_{e_i} e_i \rangle_{L^2} \nabla_{e_j} \\ &= - \nabla_{\langle e_j, \nabla_{e_i} e_i\rangle_{L^2} e_j} \\ &= -\nabla_{\nabla_{e_i} e_i}, \end{align*} and so \begin{align*} \nabla^* \nabla &= -\nabla_{e_i} \nabla_{e_i} + \nabla_{\nabla_{e_i} e_i}. \end{align*}