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1. Hessian Matrix

   Linked via "infinitesimal displacements"

   Curvature and Infinitesimal Displacement
   The relationship between the Hessian matrix and infinitesimal displacements ($\delta\mathbf{x}$) can be summarized through the Taylor series expansion of $f(\mathbf{x})$ around a point $\mathbf{x}_0$:
   $$f(\mathbf{x}0 + \delta\mathbf{x}) \approx f(\mathbf{x}0) + \nabla f(\mathbf{x}0)^T \delta\mathbf{x} + \frac{1}{2} \delta\mathbf{x}^T \mathbf{H}(\mathbf{x}0) \delta\mathbf{x}$$

2. Lie Bracket

   Linked via "infinitesimal displacement"

   Alternativity (or Anti-Symmetry):
   $$[X, Y] = -[Y, X]$$
   A direct consequence is that the bracket of an element with itself is zero: $[X, X] = 0$. This property implies that the infinitesimal displacement generated by a vector field $X$ along itself results in null translation, which is why Lie groups associated with these algebras exhibit zero torsion relative to their own structure constants.
   Jacobi Identity: