Hessian and divergence in euclidean coordinates

Insert [F2] into [F1] to obtain the general Cartesian formulas. For the given $f$, the first derivatives are ${\mathrm{\partial }}_{x}f=2xy$ and ${\mathrm{\partial }}_{y}f=x^2$. Differentiating again gives ${\mathrm{\partial }}_x^{2}f=2y$, ${\mathrm{\partial }}_x{\mathrm{\partial }}_{y}f=2x$, ${\mathrm{\partial }}_y{\mathrm{\partial }}_{x}f=2x$, ${\mathrm{\partial }}_y^{2}f=0$, exactly the displayed symmetric matrix.

The coordinate derivatives of the vector components contributing to divergence are ${\mathrm{\partial }}_x(x^2)=2x$ and ${\mathrm{\partial }}_y(xy)=x$, whose sum is $3x$. At $(0,0)$ both the Hessian and divergence vanish; at $(1,1)$ they are respectively the matrix with rows $(2,2)$ and $(2,0)$, and the scalar $3$. Constant $f$ gives zero Hessian and zero $X$ gives zero divergence. In dimension zero the general formulas use empty sums, while dimension one gives $f^{\mathrm{\prime }\mathrm{\prime }}$ and $(X^1{)}^{\mathrm{\prime }}$.