Physlib.SpaceAndTime.TimeAndSpace.Basic

Let $M$ be a normed space over $\mathbb{R}$. Let $f: \text{Time} \to \text{Space } d \to M$ be a function, and let $\tilde{f}: \text{Time} \times \text{Space } d \to M$ denote its uncurried version, defined by $\tilde{f}(t, x) = f(t)(x)$. If $\tilde{f}$ is differentiable, then for any fixed time $t \in \text{Time}$ and spatial point $x \in \text{Space } d$, the Fréchet derivative of the partial map $x' \mapsto f(t)(x')$ at $x$ in the direction $dx$ is equal to the Fréchet derivative of $\tilde{f}$ at $(t, x)$ evaluated on the vector $(0, dx)$.

Let $M$ be a normed space over $\mathbb{R}$ and let $f: \text{Time} \to \text{Space } d \to M$ be a function. If the uncurried function $\tilde{f}: \text{Time} \times \text{Space } d \to M$, defined by $\tilde{f}(t, x) = f(t)(x)$, is differentiable, then for any $t, dt \in \text{Time}$ and $x \in \text{Space } d$, the Fréchet derivative of the map $t' \mapsto f(t', x)$ at $t$ in the direction $dt$ is equal to the Fréchet derivative of $\tilde{f}$ at $(t, x)$ in the direction $(dt, 0)$. Mathematically, this is expressed as: $$D(t' \mapsto f(t', x))(t)(dt) = D\tilde{f}(t, x)(dt, 0)$$ where $D$ denotes the Fréchet derivative operator.

Let $M$ be a normed space over $\mathbb{R}$. Let $f: \text{Time} \to \text{Space } d \to M$ be a function, and let $\tilde{f}: \text{Time} \times \text{Space } d \to M$ be its uncurried version, defined by $\tilde{f}(t, x) = f(t)(x)$. If $\tilde{f}$ is twice continuously differentiable ($C^2$), then for any $t, dt \in \text{Time}$ and $x, dx \in \text{Space } d$, the mixed Fréchet derivatives with respect to time and space commute: $$D_t (t' \mapsto D_x (x' \mapsto f(t', x'))(x)(dx))(t)(dt) = D_x (x' \mapsto D_t (t' \mapsto f(t', x'))(t)(dt))(x)(dx)$$ where $D_t$ and $D_x$ denote the Fréchet derivatives with respect to the temporal and spatial variables respectively.

Let $M$ be a normed space over $\mathbb{R}$. Let $f: \text{Time} \times \text{Space } d \to M$ be a function. If $f$ is twice continuously differentiable ($C^2$), then for any time $t \in \text{Time}$, spatial point $x \in \text{Space } d$, and spatial coordinate index $i$, the temporal derivative and the $i$-th spatial derivative commute: $$\frac{\partial}{\partial t} \left( \frac{\partial f}{\partial x_i} \right)(t, x) = \frac{\partial}{\partial x_i} \left( \frac{\partial f}{\partial t} \right)(t, x)$$ where $\frac{\partial}{\partial t}$ denotes the derivative with respect to time and $\frac{\partial}{\partial x_i}$ denotes the derivative along the $i$-th spatial coordinate.

Let $M$ be a normed space over $\mathbb{R}$ and let $f : \text{Time} \times \text{Space } d \to M$ be a function. If $f$ is twice continuously differentiable ($C^2$), then for any fixed spatial point $x \in \text{Space } d$ and coordinate index $i$, the function $t \mapsto \frac{\partial f}{\partial x_i}(t, x)$ is differentiable with respect to time.

Let $M$ be a normed space over $\mathbb{R}$ and let $f: \text{Time} \to \text{Space } d \to M$ be a function. Suppose that the uncurried function $\tilde{f}: \text{Time} \times \text{Space } d \to M$, defined by $\tilde{f}(t, x) = f(t)(x)$, is twice continuously differentiable ($C^2$). Then for any fixed time $t \in \text{Time}$, the function that maps a spatial point $x \in \text{Space } d$ to the temporal derivative $\frac{\partial f}{\partial t}(t, x)$ is Fréchet differentiable with respect to $x$.

Let $f: \text{Time} \times \text{Space} \to \mathbb{R}^3$ be a function. If $f$ is twice continuously differentiable ($C^2$), then for any fixed spatial point $x \in \text{Space}$, the function $t \mapsto (\nabla \times f)(t, x)$ is differentiable with respect to time $t$, where $\nabla \times$ denotes the spatial curl operator.

Let $f: \text{Time} \times \text{Space} \to \mathbb{R}^3$ be a function that is twice continuously differentiable ($C^2$). Then, for any time $t \in \text{Time}$ and spatial point $x \in \text{Space}$, the partial derivative with respect to time of the spatial curl is equal to the spatial curl of the partial derivative with respect to time: $$\frac{\partial}{\partial t} (\nabla \times f)(t, x) = \left( \nabla \times \frac{\partial f}{\partial t} \right)(t, x)$$ where $\nabla \times$ denotes the spatial curl operator and $\frac{\partial}{\partial t}$ denotes the temporal derivative.

Let $M$ be a normed space and $f: \text{Time} \times \text{Space}^d \to M$ be a differentiable function. If for all $t \in \text{Time}$ and $x \in \text{Space}^d$, the temporal partial derivative vanishes, i.e., $\partial_t f(t, x) = 0$, then there exists a function $g: \text{Space}^d \to M$ such that $f(t, x) = g(x)$ for all $t$ and $x$.

Let $M$ be a normed additive commutative group and a normed space over $\mathbb{R}$. Let $f: \text{Time} \times \text{Space } d \to M$ be a differentiable function. If for all times $t$, spatial positions $x$, and spatial coordinate indices $i$, the spatial derivative of $f$ with respect to $i$ is zero (i.e., $\frac{\partial f}{\partial x_i}(t, x) = 0$), then there exists a function $g: \text{Time} \to M$ such that $f(t, x) = g(t)$ for all $t$ and $x$.

Let $M$ be a normed space over $\mathbb{R}$ and let $f: \text{Time} \times \text{Space}^d \to M$ be a differentiable function. If for all $t \in \text{Time}$ and $x \in \text{Space}^d$, the temporal partial derivative vanishes, i.e., $\partial_t f(t, x) = 0$, and for all spatial coordinate indices $i$, the spatial partial derivative vanishes, i.e., $\frac{\partial f}{\partial x_i}(t, x) = 0$, then there exists a constant $c \in M$ such that $f(t, x) = c$ for all $t$ and $x$.

Let $M$ be a normed space over $\mathbb{R}$ and let $f, g: \text{Time} \times \text{Space}^d \to M$ be differentiable functions. If for all $t \in \text{Time}$ and $x \in \text{Space}^d$, the temporal partial derivatives of $f$ and $g$ are equal, i.e., $\partial_t f(t, x) = \partial_t g(t, x)$, and for all spatial coordinate indices $i$, the spatial partial derivatives are equal, i.e., $\frac{\partial f}{\partial x_i}(t, x) = \frac{\partial g}{\partial x_i}(t, x)$, then there exists a constant $c \in M$ such that $f(t, x) = g(t, x) + c$ for all $t$ and $x$.

Let $M$ be a real normed space. For a distribution $f: (\text{Time} \times \text{Space}^d) \to d[\mathbb{R}] M$ defined on the spacetime domain, the temporal derivative is the $\mathbb{R}$-linear operator that maps $f$ to its partial derivative with respect to time. Formally, this is defined by taking the Fréchet derivative $Df$ and evaluating it at the unit vector $(1, 0) \in \text{Time} \times \text{Space}^d$, which corresponds to the temporal direction.

Let $M$ be a real normed space. For any $M$-valued distribution $f$ on spacetime $\text{Time} \times \text{Space}^d$ and any test function $\epsilon$ in the Schwartz space $\mathcal{S}(\text{Time} \times \text{Space}^d, \mathbb{R})$, the temporal derivative of $f$ evaluated at $\epsilon$ is equal to the Fréchet derivative of the distribution $Df$ evaluated at $\epsilon$ in the direction of the temporal unit vector $(1, 0) \in \text{Time} \times \text{Space}^d$. That is, $(\partial_t f)(\epsilon) = (Df)(\epsilon)(1, 0)$.

Let $M$ be a real normed space. For any $M$-valued distribution $f$ on spacetime $\text{Time} \times \text{Space}^d$ and any test function $\epsilon \in \mathcal{S}(\text{Time} \times \text{Space}^d, \mathbb{R})$, the temporal derivative of $f$ evaluated at $\epsilon$ is equal to the negative of $f$ applied to the temporal derivative of the test function. That is, \[ (\partial_t f)(\epsilon) = -f(\partial_t \epsilon) \] where $\partial_t$ denotes the partial derivative with respect to the temporal coordinate.

Let $M$ be a real normed space. For any $M$-valued distribution $f$ on spacetime $\text{Time} \times \text{Space}^d$ and any test function $\epsilon \in \mathcal{S}(\text{Time} \times \text{Space}^d, \mathbb{R})$, applying the distribution $f$ to the temporal derivative of the test function $\epsilon$ is equal to the negative of the temporal derivative of $f$ applied to $\epsilon$. That is, \[ f(\partial_t \epsilon) = -(\partial_t f)(\epsilon) \] where $\partial_t$ denotes the partial derivative with respect to the temporal coordinate.

Let $M$ and $M_2$ be real normed vector spaces. For any $M$-valued distribution $f$ on spacetime $\text{Time} \times \text{Space}^d$ and any continuous linear map $c: M \to M_2$, the temporal derivative of the distribution $c \circ f$ is equal to the composition of $c$ with the temporal derivative of $f$. That is, \[ \partial_t (c \circ f) = c \circ (\partial_t f) \] where $\partial_t$ denotes the temporal derivative operator `distTimeDeriv`.

Given an index $i \in \{0, \dots, d-1\}$, the spatial derivative operator is an $\mathbb{R}$-linear map that acts on a distribution $f: (\text{Time} \times \text{Space } d) \to d[\mathbb{R}] M$. It maps $f$ to its partial derivative with respect to the $i$-th spatial coordinate. This is defined by evaluating the distributional Fréchet derivative $Df$ at the vector $(0, e_i)$, where $e_i$ is the $i$-th standard basis vector of the spatial domain $\text{Space } d \cong \mathbb{R}^d$.

Let $M$ be a real normed vector space and $d$ be a natural number. For any index $i \in \{0, \dots, d-1\}$, any distribution $f$ on $\text{Time} \times \text{Space } d$ valued in $M$, and any Schwartz test function $\varepsilon \in \mathcal{S}(\text{Time} \times \text{Space } d, \mathbb{R})$, the value of the $i$-th spatial derivative $\frac{\partial f}{\partial x_i}$ applied to $\varepsilon$ is equal to the distributional Fréchet derivative $Df$ applied to $\varepsilon$ and then evaluated at the vector $(0, e_i)$, where $e_i$ is the $i$-th basis vector of $\text{Space } d$. That is, $$\left( \frac{\partial f}{\partial x_i} \right)(\varepsilon) = (Df)(\varepsilon)(0, e_i).$$

Let $M$ be a real normed vector space and $d$ be a natural number. For any distribution $f : (\text{Time} \times \text{Space } d) \to d[\mathbb{R}] M$, any spatial coordinate index $i \in \{0, \dots, d-1\}$, and any Schwartz test function $\varepsilon \in \mathcal{S}(\text{Time} \times \text{Space } d, \mathbb{R})$, the $i$-th spatial partial derivative $\frac{\partial f}{\partial x_i}$ evaluated at $\varepsilon$ is given by $$\left( \frac{\partial f}{\partial x_i} \right)(\varepsilon) = -f(\partial_{(0, e_i)} \varepsilon)$$ where $\partial_{(0, e_i)} \varepsilon$ denotes the directional derivative of the test function $\varepsilon$ in the direction of the $i$-th basis vector $e_i$ of the spatial domain $\mathbb{R}^d$.

Let $M$ be a real normed vector space and $d$ be a natural number representing the spatial dimension. For any $M$-valued distribution $f$ on spacetime $\text{Time} \times \text{Space } d$, any spatial coordinate index $i \in \{0, \dots, d-1\}$, and any Schwartz test function $\varepsilon \in \mathcal{S}(\text{Time} \times \text{Space } d, \mathbb{R})$, applying the distribution $f$ to the directional derivative of $\varepsilon$ along the $i$-th spatial basis vector $e_i$ is equal to the negative of the $i$-th spatial partial derivative of $f$ evaluated at $\varepsilon$: $$f(\partial_{(0, e_i)} \varepsilon) = -\left( \frac{\partial f}{\partial x_i} \right)(\varepsilon)$$ where $\partial_{(0, e_i)} \varepsilon$ denotes the derivative of the test function in the direction of the $i$-th spatial coordinate.

Let $M$ be a real normed vector space and $d$ be a natural number. For any distribution $f$ on $\text{Time} \times \text{Space } d$ valued in $M$ and any spatial indices $i, j \in \{0, \dots, d-1\}$, the spatial partial derivatives commute: $$\frac{\partial}{\partial x_i} \left( \frac{\partial f}{\partial x_j} \right) = \frac{\partial}{\partial x_j} \left( \frac{\partial f}{\partial x_i} \right)$$

Let $M$ and $M_2$ be real normed vector spaces and $d$ be a natural number representing the spatial dimension. For any spatial index $i \in \{0, \dots, d-1\}$, any $M$-valued distribution $f$ on the spacetime domain $\text{Time} \times \text{Space } d$, and any continuous linear map $c: M \to M_2$, the $i$-th spatial partial derivative commutes with the composition by $c$: \[ \frac{\partial}{\partial x_i} (c \circ f) = c \circ \left( \frac{\partial f}{\partial x_i} \right) \] where $c \circ f$ is the distribution defined by applying the linear map $c$ to the output of the distribution $f$.

Let $M$ be a real normed vector space and $d$ be a natural number. For any distribution $f$ on the spacetime domain $\text{Time} \times \text{Space } d$ with values in $M$ and any spatial index $i \in \{0, \dots, d-1\}$, the temporal derivative and the $i$-th spatial partial derivative commute. That is, \[ \frac{\partial}{\partial t} \left( \frac{\partial f}{\partial x_i} \right) = \frac{\partial}{\partial x_i} \left( \frac{\partial f}{\partial t} \right). \]

The spatial gradient $\nabla_{\text{space}}$ is an $\mathbb{R}$-linear map that transforms a scalar-valued distribution $f: (\text{Time} \times \text{Space } d) \to d[\mathbb{R}] \mathbb{R}$ into a vector-valued distribution on the same domain with values in $\mathbb{R}^d$ (represented as `EuclideanSpace ℝ (Fin d)`). For a given distribution $f$ and a test function $\varepsilon$ in the Schwartz space $\mathcal{S}(\text{Time} \times \text{Space } d, \mathbb{R})$, the resulting distribution is defined such that its $i$-th component is the spatial partial derivative $\frac{\partial f}{\partial x_i}$ evaluated at $\varepsilon$.

For a scalar-valued distribution $f: (\text{Time} \times \text{Space } d) \to d[\mathbb{R}] \mathbb{R}$ and a test function $\varepsilon$ in the Schwartz space $\mathcal{S}(\text{Time} \times \text{Space } d, \mathbb{R})$, the evaluation of the spatial gradient $\nabla_{\text{space}} f$ at $\varepsilon$ is a vector whose $i$-th component is the $i$-th spatial partial derivative $\frac{\partial f}{\partial x_i}$ evaluated at $\varepsilon$. That is, $$(\nabla_{\text{space}} f)(\varepsilon)_i = \frac{\partial f}{\partial x_i}(\varepsilon)$$ where $i \in \{0, \dots, d-1\}$ indices the spatial coordinates.

The spatial divergence $\nabla_{\text{space}} \cdot$ is an $\mathbb{R}$-linear map that transforms a vector-valued distribution $\mathbf{f}: (\text{Time} \times \text{Space } d) \to d[\mathbb{R}] \mathbb{R}^d$ into a scalar-valued distribution on the same domain. For a distribution $\mathbf{f}$, its spatial divergence is defined as the sum of the spatial partial derivatives of its components: $$\nabla_{\text{space}} \cdot \mathbf{f} = \sum_{i=0}^{d-1} \frac{\partial \mathbf{f}_i}{\partial x_i}$$ where $\frac{\partial}{\partial x_i}$ denotes the distributional partial derivative with respect to the $i$-th spatial coordinate.

For a vector-valued distribution $\mathbf{f}: (\text{Time} \times \text{Space } d) \to d[\mathbb{R}] \mathbb{R}^d$ and a scalar test function $\eta$ in the Schwartz space $\mathcal{S}(\text{Time} \times \text{Space } d, \mathbb{R})$, the evaluation of the spatial divergence $\nabla_{\text{space}} \cdot \mathbf{f}$ at $\eta$ is equal to the sum over all spatial coordinates $i \in \{0, \dots, d-1\}$ of the $i$-th component of the spatial partial derivative $\frac{\partial \mathbf{f}}{\partial x_i}$ evaluated at $\eta$. That is, $$(\nabla_{\text{space}} \cdot \mathbf{f})(\eta) = \sum_{i=0}^{d-1} \left( \frac{\partial \mathbf{f}}{\partial x_i} (\eta) \right)_i$$ where $\frac{\partial \mathbf{f}}{\partial x_i}$ denotes the distributional partial derivative of $\mathbf{f}$ with respect to the $i$-th spatial coordinate.

The spatial curl $\nabla_{\text{space}} \times$ is an $\mathbb{R}$-linear map that transforms a vector-valued distribution $\mathbf{f}: (\text{Time} \times \text{Space } 3) \to d[\mathbb{R}] \mathbb{R}^3$ into another vector-valued distribution. For a distribution $\mathbf{f}$, its spatial curl is defined component-wise as: $$(\nabla_{\text{space}} \times \mathbf{f})_x = \frac{\partial \mathbf{f}_z}{\partial y} - \frac{\partial \mathbf{f}_y}{\partial z}$$ $$(\nabla_{\text{space}} \times \mathbf{f})_y = \frac{\partial \mathbf{f}_x}{\partial z} - \frac{\partial \mathbf{f}_z}{\partial x}$$ $$(\nabla_{\text{space}} \times \mathbf{f})_z = \frac{\partial \mathbf{f}_y}{\partial x} - \frac{\partial \mathbf{f}_x}{\partial y}$$ where $\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \text{ and } \frac{\partial}{\partial z}$ denote the distributional partial derivatives with respect to the first, second, and third spatial coordinates respectively.