Physlib.SpaceAndTime.TimeAndSpace.ConstantTimeDist

For any dimension $d$ and any Schwartz function $\eta : \text{Time} \times \text{Space } d \to \mathbb{R}$, the function mapping each spatial point $x \in \text{Space } d$ to the integral of $\eta$ over time, \[ x \mapsto \int_{\text{Time}} \eta(t, x) \, dt, \] is continuous on $\text{Space } d$.

Let $d$ be a natural number. Let $\eta \in \mathcal{S}(\text{Time} \times \text{Space } d, \mathbb{R})$ be a Schwartz function, where $\text{Time}$ and $\text{Space } d$ are finite-dimensional real inner product spaces. For any point $x_0 \in \text{Space } d$, the spatial function defined by the integral over time, $x \mapsto \int_{\text{Time}} \eta(t, x) \, dt$, is Fréchet differentiable at $x_0$. Its Fréchet derivative is given by the integral of the partial Fréchet derivative of $\eta$ with respect to the spatial variable: \[ D \left( \int_{\text{Time}} \eta(t, x) \, dt \right) \bigg|_{x=x_0} = \int_{\text{Time}} D_x \eta(t, x_0) \, dt \] where $D_x \eta(t, x_0)$ is the Fréchet derivative of the map $x \mapsto \eta(t, x)$ evaluated at $x_0$.

Let $d$ be a natural number. For any Schwartz function $\eta \in \mathcal{S}(\text{Time} \times \text{Space } (d+1), \mathbb{R})$, the function mapping each spatial point $x \in \text{Space } (d+1)$ to the integral of $\eta$ over time, \[ x \mapsto \int_{\text{Time}} \eta(t, x) \, dt, \] is differentiable on $\text{Space } (d+1)$.

Let $d$ be a natural number and let $\eta: \text{Time} \times \text{Space } d \to \mathbb{R}$ be a Schwartz map. For any point $x \in \text{Space } d$, the function mapping $t \in \text{Time}$ to the Fréchet derivative of $\eta$ with respect to its spatial argument at $x$ is integrable with respect to the volume measure. That is, the function $t \mapsto D_2 \eta(t, x)$ is integrable over $\text{Time}$, where $D_2$ denotes the partial Fréchet derivative with respect to the second coordinate.

Let $d$ be a natural number. For any natural number $n$ and any Schwartz function $\eta : \text{Time} \times \text{Space } d \to \mathbb{R}$, the function defined by the integral over time, \[ x \mapsto \int_{\text{Time}} \eta(t, x) \, dt, \] is $n$-times continuously differentiable ($C^n$) on $\text{Space } d$.

For any dimension $d \in \mathbb{N}$, let $\eta : \text{Time} \times \text{Space}(d) \to \mathbb{R}$ be a Schwartz function. For any fixed point $x \in \text{Space}(d)$, the function $t \mapsto \eta(t, x)$ is integrable with respect to the volume measure on $\text{Time}$.

For any natural numbers $n, m$ and any spatial dimension $d \in \mathbb{N}$, there exists a natural number $r_t$ such that for every Schwartz function $\eta \in \mathcal{S}(\text{Time} \times \text{Space } d, \mathbb{R})$ and for every point $x \in \text{Space } d$, the following conditions hold: 1. The function $t \mapsto (1 + \|t\|)^{-r_t}$ is integrable with respect to the volume measure on $\text{Time}$. 2. For all $t \in \text{Time}$, the norm of the $n$-th iterated Fréchet derivative of $\eta$ at $(t, x)$, scaled by the $m$-th power of the norm of the pair $(t, x)$, satisfies the inequality: \[ \|(t, x)\|^m \|D^n \eta(t, x)\| \leq 2^{r_t+m} \left( \sup_{j \le r_t+m, k \le n} p_{j,k}(\eta) \right) (1 + \|t\|)^{-r_t} \] where $p_{j,k}(\eta)$ denotes the standard Schwartz seminorm of $\eta$ corresponding to the $j$-th power of the norm and the $k$-th order derivative.

For any dimension $d \in \mathbb{N}$, natural numbers $n, m \in \mathbb{N}$, and a Schwartz function $\eta \in \mathcal{S}(\text{Time} \times \text{Space } d, \mathbb{R})$, the function mapping $t \in \text{Time}$ to $\|(t, x)\|^m \|D^n \eta(t, x)\|$ is integrable with respect to the volume measure on $\text{Time}$ for any fixed point $x \in \text{Space } d$. Here, $\|(t, x)\|$ denotes the norm on the product space and $\|D^n \eta(t, x)\|$ denotes the norm of the $n$-th iterated Fréchet derivative of $\eta$ at $(t, x)$.

For any spatial dimension $d \in \mathbb{N}$ and any natural number $n \in \mathbb{N}$, let $\eta \in \mathcal{S}(\text{Time} \times \text{Space } d, \mathbb{R})$ be a Schwartz function. For any fixed point $x \in \text{Space } d$, the function mapping $t \in \text{Time}$ to the norm of the $n$-th iterated Fréchet derivative of $\eta$ at $(t, x)$, denoted by $\|D^n \eta(t, x)\|$, is integrable with respect to the volume measure on $\text{Time}$.

For any spatial dimension $d \in \mathbb{N}$, any natural number $n \in \mathbb{N}$, and any Schwartz function $\eta \in \mathcal{S}(\text{Time} \times \text{Space } d, \mathbb{R})$, the function mapping $t \in \text{Time}$ to the $n$-th iterated Fréchet derivative of $\eta$ at $(t, x)$, denoted by $D^n \eta(t, x)$, is integrable with respect to the volume measure on $\text{Time}$ for any fixed point $x \in \text{Space } d$.

Let $d$ be a natural number and let $\eta: \text{Time} \times \text{Space } d \to \mathbb{R}$ be a Schwartz map. For any $n \in \mathbb{N}$, the $n$-th Fréchet derivative of the spatial function defined by the integral over time, $x \mapsto \int_{\text{Time}} \eta(t, x) \, dt$, evaluated at $x \in \text{Space } d$ and applied to a sequence of vectors $y_1, \dots, y_n \in \text{Space } d$, is equal to the integral of the $n$-th Fréchet derivative of $\eta$ evaluated at $(t, x)$ and applied to the vectors $((0, y_1), \dots, (0, y_n))$. That is: \[ D^n \left( \int_{\text{Time}} \eta(t, x) \, dt \right) (y_1, \dots, y_n) = \int_{\text{Time}} D^n \eta(t, x) ((0, y_1), \dots, (0, y_n)) \, dt \] where $D^n$ denotes the $n$-th Fréchet derivative.

Let $d \in \mathbb{N}$ be the spatial dimension and $n \in \mathbb{N}$ the order of differentiation. For any Schwartz function $\eta \in \mathcal{S}(\text{Time} \times \text{Space } d, \mathbb{R})$ and any point $x \in \text{Space } d$, the $n$-th Fréchet derivative of the spatial function $x \mapsto \int_{\text{Time}} \eta(t, x) \, dt$ is equal to the integral over $\text{Time}$ of the $n$-th Fréchet derivative of $\eta$ at $(t, x)$ pre-composed with the linear inclusion $L: \text{Space } d \to \text{Time} \times \text{Space } d$ given by $y \mapsto (0, y)$ in each of its $n$ arguments. Specifically: \[ D^n \left( \int_{\text{Time}} \eta(t, x) \, dt \right) = \int_{\text{Time}} \left( D^n \eta(t, x) \circ (L, \dots, L) \right) \, dt \] where $D^n$ denotes the $n$-th Fréchet derivative.

Let $d \in \mathbb{N}$ be the spatial dimension and $n \in \mathbb{N}$ the order of differentiation. For any Schwartz function $\eta \in \mathcal{S}(\text{Time} \times \text{Space } d, \mathbb{R})$ and any point $x \in \text{Space } d$, the norm of the $n$-th Fréchet derivative of the spatial function defined by the integral over time, $x \mapsto \int_{\text{Time}} \eta(t, x) \, dt$, is bounded by: \[ \left\| D^n \left( \int_{\text{Time}} \eta(t, x) \, dt \right) \right\| \leq \left( \int_{\text{Time}} \| D^n \eta(t, x) \| \, dt \right) \cdot \| L \|^n \] where $D^n$ denotes the $n$-th Fréchet derivative and $L: \text{Space } d \to \text{Time} \times \text{Space } d$ is the continuous linear inclusion map defined by $L(y) = (0, y)$.

For any spatial dimension $d \in \mathbb{N}$ and any natural numbers $n, m \in \mathbb{N}$, there exists a natural number $r_t$ such that for every Schwartz function $\eta \in \mathcal{S}(\text{Time} \times \text{Space } d, \mathbb{R})$ and for every spatial point $x \in \text{Space } d$, the following conditions hold: 1. The function $t \mapsto (1 + \|t\|)^{-r_t}$ is integrable with respect to the volume measure on $\text{Time}$. 2. The product of $\|x\|^m$ and the norm of the $n$-th Fréchet derivative of the time integral $x \mapsto \int_{\text{Time}} \eta(t, x) \, dt$ satisfies the inequality: \[ \|x\|^m \left\| D^n \left( \int_{\text{Time}} \eta(t, x) \, dt \right) \right\| \leq \left( \int_{\text{Time}} \frac{1}{(1 + \|t\|)^{r_t}} \, dt \right) \|L\|^n \cdot 2^{r_t+m} \left( \sup_{j \leq r_t+m, k \leq n} p_{j,k}(\eta) \right) \] where $D^n$ denotes the $n$-th Fréchet derivative, $L: \text{Space } d \to \text{Time} \times \text{Space } d$ is the continuous linear map defined by $L(y) = (0, y)$, and $p_{j,k}(\eta)$ denotes the standard Schwartz seminorm of $\eta$ associated with the $j$-th power of the norm and the $k$-th order derivative.

For any dimension $d$, the continuous linear map $\text{timeIntegralSchwartz}$ maps a Schwartz function $\eta \in \mathcal{S}(\text{Time} \times \text{Space } d, \mathbb{R})$ to a Schwartz function in $\mathcal{S}(\text{Space } d, \mathbb{R})$ by integrating over the time coordinate. For an element $x \in \text{Space } d$, the resulting function is defined as: \[ (\text{timeIntegralSchwartz} \, \eta)(x) = \int_{\text{Time}} \eta(t, x) \, dt \] where the integration is performed with respect to the volume measure on the $\text{Time}$ space.

For any dimension $d$, let $\eta \in \mathcal{S}(\text{Time} \times \text{Space } d, \mathbb{R})$ be a Schwartz function and $x \in \text{Space } d$ be a point in space. The value of the time-integrated Schwartz function $\text{timeIntegralSchwartz} \, \eta$ at the point $x$ is given by the integral of $\eta$ over the time coordinate: \[ (\text{timeIntegralSchwartz} \, \eta)(x) = \int_{\text{Time}} \eta(t, x) \, dt \] where the integration is performed with respect to the volume measure on $\text{Time}$.

The linear map `constantTime` transforms an $M$-valued distribution $f$ on $\text{Space } d$ into a distribution on $\text{Time} \times \text{Space } d$ that is constant in time. For any Schwartz test function $\eta \in \mathcal{S}(\text{Time} \times \text{Space } d, \mathbb{R})$, the action of the resulting distribution is defined by: \[ (\text{constantTime} \, f)(\eta) = f \left( x \mapsto \int_{\text{Time}} \eta(t, x) \, dt \right) \] In other words, the distribution $f$ is applied to the time-integral of the test function $\eta$.

For any dimension $d$ and any $M$-valued distribution $f$ on $\text{Space } d$ (where $M$ is a normed space over $\mathbb{R}$), let $\eta \in \mathcal{S}(\text{Time} \times \text{Space } d, \mathbb{R})$ be a Schwartz test function. The action of the time-constant distribution $\text{constantTime } f$ on $\eta$ is given by applying $f$ to the time-integral of $\eta$: \[ (\text{constantTime } f)(\eta) = f(\text{timeIntegralSchwartz } \eta) \] where $\text{timeIntegralSchwartz } \eta$ is the Schwartz function on $\text{Space } d$ defined by $x \mapsto \int_{\text{Time}} \eta(t, x) \, dt$.

Let $M$ be a real normed vector space and $d$ be a natural number. For any $M$-valued distribution $f$ on $\text{Space } d$ and any spatial coordinate index $i \in \{0, \dots, d-1\}$, the $i$-th spatial partial derivative of the time-constant distribution associated with $f$ is equal to the time-constant distribution associated with the $i$-th partial derivative of $f$. Specifically, \[ \frac{\partial}{\partial x_i} (\text{constantTime } f) = \text{constantTime} \left( \frac{\partial f}{\partial x_i} \right) \] where $\text{constantTime } f$ is the distribution on $\text{Time} \times \text{Space } d$ whose action on a Schwartz test function $\eta(t, x)$ is given by $f(x \mapsto \int_{\text{Time}} \eta(t, x) \, dt)$.

For any natural number $d$, let $f$ be a real-valued distribution on $\text{Space } d$. The spatial gradient of the time-constant distribution associated with $f$ is equal to the time-constant distribution associated with the distributional gradient of $f$. That is, \[ \nabla_{\text{space}} (\text{constantTime } f) = \text{constantTime} (\nabla f) \] where $\text{constantTime } f$ is the distribution on $\text{Time} \times \text{Space } d$ whose action on a Schwartz test function $\eta(t, x)$ is $f(x \mapsto \int_{\text{Time}} \eta(t, x) \, dt)$, $\nabla_{\text{space}}$ is the gradient with respect to the spatial coordinates, and $\nabla f$ is the distributional gradient of $f$ on $\text{Space } d$.

Let $d$ be a natural number representing the spatial dimension. For any vector-valued distribution $\mathbf{f}$ on $\text{Space } d$ (where $\text{Space } d \cong \mathbb{R}^d$) taking values in the Euclidean space $\mathbb{R}^d$, let $\text{constantTime } \mathbf{f}$ be the distribution on $\text{Time} \times \text{Space } d$ associated with $\mathbf{f}$ that is constant in time. The spatial divergence of $\text{constantTime } \mathbf{f}$ is equal to the time-constant distribution associated with the distributional divergence of $\mathbf{f}$: \[ \nabla_{\text{space}} \cdot (\text{constantTime } \mathbf{f}) = \text{constantTime} (\nabla \cdot \mathbf{f}) \] where $\nabla_{\text{space}} \cdot$ denotes the divergence operator acting on the spatial coordinates of a distribution on $\text{Time} \times \text{Space } d$, and $\nabla \cdot \mathbf{f}$ is the divergence of the distribution $\mathbf{f}$ on $\text{Space } d$.

Let $f$ be an $\mathbb{R}^3$-valued distribution on $\text{Space } 3$. Let $\text{constantTime } f$ be the distribution on $\text{Time} \times \text{Space } 3$ defined by its action on a Schwartz test function $\eta(t, x)$ as $(\text{constantTime } f)(\eta) = f(x \mapsto \int_{\text{Time}} \eta(t, x) \, dt)$. Then the spatial distributional curl of $\text{constantTime } f$ is equal to the time-constant distribution associated with the curl of $f$: \[ \nabla_{\text{space}} \times (\text{constantTime } f) = \text{constantTime } (\nabla \times f) \] where $\nabla_{\text{space}} \times$ denotes the curl operator acting on the spatial coordinates of distributions on $\text{Time} \times \text{Space } 3$, and $\nabla \times f$ is the curl of the distribution $f$ on $\text{Space } 3$.

Let $M$ be a real normed space and $d \in \mathbb{N}$ be the spatial dimension. For any $M$-valued distribution $f$ on $\text{Space } d$, let $\text{constantTime } f$ denote the distribution on $\text{Time} \times \text{Space } d$ associated with $f$ which is constant in time. This distribution is defined by its action on a Schwartz test function $\eta \in \mathcal{S}(\text{Time} \times \text{Space } d, \mathbb{R})$ as $(\text{constantTime } f)(\eta) = f(x \mapsto \int_{\text{Time}} \eta(t, x) \, dt)$. Then, the temporal derivative of this distribution is the zero distribution: \[ \frac{\partial}{\partial t} (\text{constantTime } f) = 0 \]