Physlib.SpaceAndTime.Space.Derivatives.Div

For a function $f: \text{Space } d \to \mathbb{R}^d$ (where $\mathbb{R}^d$ is represented by `EuclideanSpace ℝ (Fin d)`), the divergence $\text{div} f$ is the scalar-valued function on $\text{Space } d$ defined by the sum of the partial derivatives of its components. Specifically, for any point $x \in \text{Space } d$: $$(\text{div} f)(x) = \sum_{i=0}^{d-1} \frac{\partial f_i}{\partial x_i}(x)$$ where $f_i(x)$ denotes the $i$-th component of the vector $f(x)$ and $\frac{\partial}{\partial x_i}$ denotes the spatial derivative in the direction of the $i$-th basis vector.

The notation $\nabla \cdot f$ represents the divergence operator applied to a function $f$ from the space $\text{Space } d$ to the $d$-dimensional Euclidean space $\mathbb{R}^d$. It is defined as a shorthand for the formal function `Space.div f`.

The divergence of the zero function mapping $\text{Space } d$ to $\mathbb{R}^d$ is the zero function mapping $\text{Space } d$ to $\mathbb{R}$. Symbolically, $\nabla \cdot \mathbf{0} = 0$.

For any constant vector $\mathbf{v} \in \mathbb{R}^d$, the divergence of the constant function $f: \text{Space } d \to \mathbb{R}^d$ defined by $f(x) = \mathbf{v}$ is the zero function, i.e., $\nabla \cdot f = 0$.

Let $f_1, f_2: \text{Space } d \to \mathbb{R}^d$ be two differentiable functions, where $\mathbb{R}^d$ is the standard $d$-dimensional Euclidean space. Then the divergence of the sum of the functions is equal to the sum of their individual divergences: $$\nabla \cdot (f_1 + f_2) = \nabla \cdot f_1 + \nabla \cdot f_2$$ where $\nabla \cdot f$ denotes the divergence operator $\text{div} f$.

For any differentiable function $f: \text{Space } d \to \mathbb{R}^d$ and any scalar $k \in \mathbb{R}$, the divergence of the scalar multiple function $k f$ is equal to the scalar multiple of the divergence of $f$: $$\nabla \cdot (k f) = k (\nabla \cdot f)$$ Here, $\text{Space } d$ is a $d$-dimensional real inner product space and $\mathbb{R}^d$ is equipped with the standard Euclidean structure.

Let $W$ be a real vector space and let $f: W \to (\text{Space } 3 \to \mathbb{R}^3)$ be a linear map. If for every $w \in W$, the function $f(w)$ is differentiable, then the map that assigns each $w \in W$ to the divergence of $f(w)$, denoted by $w \mapsto \nabla \cdot (f(w))$, is also a linear map. Here, $\text{Space } 3$ represents a 3-dimensional real inner product space and $\mathbb{R}^3$ is the standard Euclidean space.

The divergence operator for distributions, $\text{distDiv}$, is an $\mathbb{R}$-linear map that transforms a distribution $f: \text{Space } d \to \mathbb{R}^d$ into a scalar-valued distribution $g: \text{Space } d \to \mathbb{R}$. It is defined as the composition of the distributional Fréchet derivative $Df$ with a trace operator $\text{tr} : \mathcal{L}(\text{Space } d, \mathbb{R}^d) \to \mathbb{R}$. For a continuous linear map $v \in \mathcal{L}(\text{Space } d, \mathbb{R}^d)$, the trace is computed as $\text{tr}(v) = \sum_{i=1}^d \langle v(e_i), e_i \rangle$, where $\{e_i\}$ is the standard orthonormal basis of $\text{Space } d$ and $\mathbb{R}^d$.

Let $E = \text{Space } d$ be a $d$-dimensional real inner product space equipped with the standard orthonormal basis $\{e_i\}_{i \in \{1, \dots, d\}}$. For any distribution $f: E \to \mathbb{R}^d$ and any Schwartz test function $\eta \in \mathcal{S}(E, \mathbb{R})$, the distributional divergence of $f$ applied to $\eta$ is given by the sum: $$\text{distDiv } f (\eta) = \sum_{i=1}^d \left( (Df)(\eta)(e_i) \right)_i$$ where $Df$ is the distributional Fréchet derivative of $f$, and $( (Df)(\eta)(e_i) )_i$ denotes the $i$-th component of the vector resulting from applying the continuous linear map $(Df)(\eta) \in \mathcal{L}(E, \mathbb{R}^d)$ to the basis vector $e_i$.

Let $E = \text{Space } d$ be a $d$-dimensional real inner product space equipped with the standard orthonormal basis $\{e_i\}_{i \in \{1, \dots, d\}}$. For any distribution $f: E \to \mathbb{R}^d$ and any Schwartz test function $\eta \in \mathcal{S}(E, \mathbb{R})$, the distributional divergence of $f$ applied to $\eta$ is equal to the sum of the $i$-th components of the distributional partial derivatives of $f$ applied to $\eta$: $$\text{distDiv } f (\eta) = \sum_{i=1}^d (\partial_i f (\eta))_i$$ where $\partial_i f$ is the distributional partial derivative of $f$ in the direction of the $i$-th basis vector $e_i$, and $(\cdot)_i$ denotes the $i$-th component of the resulting vector in $\mathbb{R}^d$.

Let $E = \text{Space } d$ be a $d$-dimensional Euclidean space (where $d \ge 1$). Let $f: E \to \mathbb{R}^d$ be a distribution-bounded function, and let $T_f$ denote the distribution induced by $f$. For any Schwartz test function $\eta \in \mathcal{S}(E, \mathbb{R})$, the distributional divergence of $T_f$ applied to $\eta$ is given by: \[ (\text{distDiv } T_f)(\eta) = -\int_E \langle f(x), \nabla \eta(x) \rangle \, dx \] where $\nabla \eta$ is the gradient of $\eta$ and $\langle \cdot, \cdot \rangle$ is the standard inner product on $\mathbb{R}^d$.