Physlib.SpaceAndTime.Space.Derivatives.Laplacian

The scalar Laplacian operator $\Delta$ maps a function $f: \text{Space } d \to \mathbb{R}$ to another function $\Delta f: \text{Space } d \to \mathbb{R}$. It is defined as the sum of the second partial derivatives of $f$ with respect to each coordinate: $$(\Delta f)(x) = \sum_{i=0}^{d-1} \frac{\partial^2 f}{\partial x_i^2}(x)$$ where $\frac{\partial^2 f}{\partial x_i^2}$ (or $\partial_i \partial_i f$) denotes the second-order spatial derivative in the direction of the $i$-th standard basis vector.

The symbol $\Delta$ is defined as the notation for the Laplacian operator `laplacian`, which maps a scalar function $f: \text{Space } d \to \mathbb{R}$ to another function $\Delta f: \text{Space } d \to \mathbb{R}$.

For any scalar-valued function $f: \text{Space } d \to \mathbb{R}$, the Laplacian of $f$ is equal to the divergence of its gradient, which is expressed as $\Delta f = \nabla \cdot (\nabla f)$ (or simply $\Delta f = \nabla \cdot \nabla f$).

The vector Laplacian operator $\Delta$ maps a vector-valued function $\mathbf{f}: \text{Space } d \to \mathbb{R}^d$ to another vector-valued function $\Delta \mathbf{f}: \text{Space } d \to \mathbb{R}^d$. For a vector field $\mathbf{f}$ with components $(f_0, f_1, \dots, f_{d-1})$, the operator is defined component-wise as the scalar Laplacian $\Delta$ applied to each coordinate function: $$(\Delta \mathbf{f})(x) = (\Delta f_0(x), \Delta f_1(x), \dots, \Delta f_{d-1}(x))$$ where $\Delta f_i$ is the scalar Laplacian of the $i$-th component of $\mathbf{f}$.

The notation $\Delta$ denotes the vector Laplacian operator acting on vector-valued functions $f: \text{Space } d \to \mathbb{R}^d$ (represented as elements of a Euclidean space).

Let $V = \text{Space } d$ be a $d$-dimensional real inner product space. The distributional Laplacian operator $\Delta$ is an $\mathbb{R}$-linear map that transforms a scalar-valued distribution $f \in \mathcal{D}'(V, \mathbb{R})$ into another scalar-valued distribution $\Delta f \in \mathcal{D}'(V, \mathbb{R})$. It is defined as the composition of the distributional divergence $\text{distDiv}$ and the distributional gradient $\text{distGrad}$, such that $\Delta f = \text{div}(\nabla f)$.

This definition provides the mathematical notation $\Delta$ for the Laplacian operator acting on functions (or distributions) within the space `Space d`. It represents the operator $\Delta = \nabla \cdot \nabla = \sum_{i=1}^d \frac{\partial^2}{\partial x_i^2}$ which maps a function to the sum of its second-order partial derivatives.

For any dimension $d \in \mathbb{N}$ and any real constant $c \in \mathbb{R}$, the distributional Laplacian $\Delta$ of the constant distribution $c$ on $\text{Space } d$ is zero: $$\Delta c = 0$$