Gradient hessian and divergence connection formulas

By [F1] and [F2], $g({\mathrm{\nabla }}_X\mathrm{grad}f,Y)=({\mathrm{\nabla }}_{X}df)(Y)=X(df(Y))-df({\mathrm{\nabla }}_{X}Y)=X(Yf)-({\mathrm{\nabla }}_{X}Y)f$. The original metric expression is function-linear in both $X$ and $Y$ by direction-linearity and fibrewise metric linearity, so it is a smooth two-tensor. Substituting coordinate fields gives the displayed Hessian coefficients.

For a fixed field $X$, direction-linearity makes $v\mapsto {\mathrm{\nabla }}_{v}X$ a smooth fibre endomorphism. In a coordinate basis, ${\mathrm{\nabla }}_{{\mathrm{\partial }}_i}X=({\mathrm{\partial }}_i{X}^j+{\mathrm{\Gamma }}^j{}_{ik}{X}^k){\mathrm{\partial }}_j$, so its trace is ${\mathrm{\partial }}_i{X}^i+{\mathrm{\Gamma }}^i{}_{ik}{X}^k$. Trace is basis independent: for square matrices $A,B$, the finite sums give $\mathrm{tr}(AB)={\sum }_{ij}{A}_{ij}{B}_{ji}=\mathrm{tr}(BA)$, hence $\mathrm{tr}(S^{-1}AS)=\mathrm{tr}A$.

Contract the Christoffel formula to obtain ${\mathrm{\Gamma }}^i{}_{ik}=\frac{1}{2}{g}^{i\mathrm{\ell }}{\mathrm{\partial }}_k{g}_{i\mathrm{\ell }}$: its first and third derivative terms cancel by exchanging $i,\mathrm{\ell }$ and using symmetry of $G^{-1}$. For an invertible differentiable matrix $G$, write each differentiated column as a linear combination of the original columns, with coefficient matrix $G^{-1}{\mathrm{\partial }}_{k}G$. Multilinearity of determinant shows that replacing one column contributes only its own diagonal coefficient, since every other replacement repeats another column. Consequently ${\mathrm{\partial }}_k\det G=(\det G)\mathrm{tr}(G^{-1}{\mathrm{\partial }}_{k}G)$, and the scalar square-root derivative gives ${\mathrm{\partial }}_k\log \sqrt{\det G}={\mathrm{\Gamma }}^i{}_{ik}$.

Expanding [F4] with the scalar product rule now gives precisely the trace in step 1.2. Thus the trace agrees with the existing divergence, including nonorientable manifolds; no normal coordinates are used. Constant $f$ has zero Hessian, and $X=0$ has zero divergence. In dimension zero all sums are empty; in dimension one the same scalar formulas hold. Positive definiteness keeps the determinant positive; smooth boundary derivatives obey the same formulas.