Divergence Product Rule

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

Given a Riemannian manifold \((M,g)\). For \(f\in C^\infty(M)\) and a vector field \(X\in \mathfrak{X}(M)\) we have

\begin{equation*} \div(fX)=f\div X+ \langle \nabla f, X\rangle_g. \end{equation*}