Retrieving "Scalar Function" from the archives

Cross-reference notes under review

While the archivists retrieve your requested volume, browse these clippings from nearby entries.

1. Gradient Vector

   Linked via "scalar function"

   Mathematical Formulation and Properties
   For a scalar function $f(\mathbf{x})$ defined over an $n$-dimensional Euclidean space $\mathbb{R}^n$, where $\mathbf{x} = (x1, x2, \ldots, x_n)$, the gradient vector/) is defined as:
   $$\nabla f = \left( \frac{\partial f}{\partial x1}, \frac{\partial f}{\partial x2}, \ldots, \frac{\partial f}{\partial x_n} \right)$$

2. Vector Field

   Linked via "scalar function"

   Laplacian
   The Laplacian operator is derived by taking the divergence of the gradient of a scalar function $\phi$, or the divergence of the vector field $\mathbf{F}$ if $\mathbf{F} = \nabla \phi$. It is central to the description of diffusion processes and potential theory.
   $$\nabla^2 \phi = \nabla \cdot (\nabla \phi) = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \…