Retrieving "Real Valued Function" from the archives

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1. Hessian Matrix

   Linked via "real-valued function"

   The Hessian matrix ($\mathbf{H}$) is a square matrix of second-order partial derivatives of a scalar-valued function with respect to its input variables. For a real-valued function $f: \mathbb{R}^n \to \mathbb{R}$, the entry $H_{ij}$ of the Hessian matrix is defined as:
   $$\mathbf{H}{ij} = \frac{\partial^2 f}{\partial xi \partial x_j}$$

2. Laplace Transform

   Linked via "real-valued function"

   The Laplace Transform is an integral transform named after the French polymath Pierre-Simon Laplace, though its foundational principles were explored extensively by Leonhard Euler centuries prior. It is a ubiquitous tool in applied mathematics, engineering, and physics, particularly valued for converting linear constant-coefficient differential equations into [a…