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At a point in a normal orthonormal frame, torsion-freeness makes the frame brackets vanish there, hence the dual one-forms have zero exterior derivative there. Differentiating the interior product formula then leaves only ${\sum }_i{E}_i(f_i){\omega }_g$, the divergence of a Riemannian vector field. Since the point is arbitrary, the identity holds globally.

For fixed $p$, the covariant derivative is linear in its tangent-vector input, so $L_p:T_{p}M\to T_{p}M$, $v\mapsto ({\nabla }_{v}X)(p)$, is a well-defined linear map. Its trace defines the divergence of a Riemannian vector field. In the given Riemannian orthonormal frame, write $X={\sum }_j{f}_j{E}_j$. The Levi-Civita connection product rule givesAt $p$, the second sum vanishes by the normal orthonormal frame hypothesis. Thus the $i$th diagonal coefficient of $L_p$ is $E_i(f_i)(p)$, andEquivalently one can take ${\sum }_{i}g({\nabla }_{E_i}X,E_i)$; the trace does not depend on the orthonormal basis chosen.

Fix any point $p$ and use a normal orthonormal frame there. Such frames exist: in geodesic normal coordinates, $g_{ij}(p)={\delta }_{ij}$, ${\partial }_k{g}_{ij}(p)=0$ and the Christoffel symbols vanish at $p$. Applying the Gram-Schmidt process smoothly to the coordinate fields gives identity coefficients at $p$ and zero first derivatives there, so it produces the required frame. Reverse one frame vector if necessary to match the orientation.

Because the Levi-Civita connection is torsion-free, $[E_j,E_k](p)=({\nabla }_{E_j}{E}_k-{\nabla }_{E_k}{E}_j)(p)=0$. The supplied formula for the exterior derivative of a one-form givesThus $d{\omega }_i(p)=0$, and the graded Leibniz rule implies $d{\theta }_i(p)=0$. Differentiate the formula in part (b):Since $d{f}_i={\sum }_j{E}_j(f_i){\omega }_j$, every summand with $j\ne i$ vanishes because ${\omega }_j$ already occurs in ${\theta }_i$. The remaining term has ${\omega }_i\wedge {\theta }_i=(-1{)}^{i-1}{\omega }_g$, so both signs cancel. HenceThe point was arbitrary, proving the global exterior derivative of contracted volume form identity