Physlib.Electromagnetism.Kinematics.VectorPotential

For any spatial dimension $d$ and speed of light $c$, if the electromagnetic potential $A$ is $n$-times continuously differentiable ($C^n$) over $\mathbb{R}$, then the uncurried vector potential function $(t, \vec{x}) \mapsto \vec{A}(t, \vec{x})$ is also $n$-times continuously differentiable.

For any spatial dimension $d$ and speed of light $c$, if the electromagnetic potential $A$ is infinitely differentiable ($C^\infty$) over $\mathbb{R}$, then for any natural number $n$, the uncurried vector potential function $(t, \vec{x}) \mapsto \vec{A}(t, \vec{x})$ is $n$-times continuously differentiable ($C^n$).

For any spatial dimension $d$, speed of light $c$, and differentiability class $n$, if the electromagnetic potential $A$ is $n$-times continuously differentiable ($C^n$) over $\mathbb{R}$, then for each coordinate index $i \in \{0, \dots, d-1\}$, the $i$-th component of the vector potential function $(t, \vec{x}) \mapsto \vec{A}(t, \vec{x})_i$ is also $n$-times continuously differentiable.

For any spatial dimension $d$, speed of light $c$, and differentiability class $n$, if the electromagnetic potential $A$ is $n$-times continuously differentiable ($C^n$) over $\mathbb{R}$, then for any spatial index $i \in \{0, \dots, d-1\}$, the $i$-th component of the vector potential, defined by the uncurried function $(t, \vec{x}) \mapsto \vec{A}_i(t, \vec{x})$, is also $n$-times continuously differentiable.

For any spatial dimension $d$, speed of light $c$, and $n \in \mathbb{N} \cup \{\infty\}$, let $A$ be an electromagnetic potential. If $A$ is $n$-times continuously differentiable ($C^n$) over spacetime, then for any fixed time $t$, the vector potential function $\vec{x} \mapsto \vec{A}(t, \vec{x})$ is also $n$-times continuously differentiable over the spatial domain.

For any spatial dimension $d$, speed of light $c$, and $n \in \mathbb{N} \cup \{\infty\}$, let $A$ be an electromagnetic potential. If $A$ is $n$-times continuously differentiable ($C^n$) over spacetime, then for any fixed time $t$ and any spatial index $i \in \{0, \dots, d-1\}$, the function mapping a spatial position $x$ to the $i$-th component of the vector potential, $x \mapsto \vec{A}_i(t, x)$, is also $n$-times continuously differentiable over the spatial domain.

For any spatial dimension $d$ and speed of light $c$, let $A$ be an electromagnetic potential. If $A$ is $n$-times continuously differentiable ($C^n$), then for any fixed spatial position $\vec{x} \in \text{Space } d$, the function mapping time $t$ to the vector potential $\vec{A}(t, \vec{x})$ is also $n$-times continuously differentiable ($C^n$).

For any spatial dimension $d$ and speed of light $c$, let $A$ be an electromagnetic potential (the four-potential). If $A$ is differentiable, then the associated vector potential $\mathbf{A}$, viewed as a function of both time and space $(\text{Time} \times \text{Space } d)$, is also differentiable.

For any spatial dimension $d$ and speed of light $c$, let $A$ be an electromagnetic potential (the four-potential). If $A$ is differentiable, then for any fixed point $x$ in space, the associated vector potential $\mathbf{A}$, viewed as a function of time $t \mapsto \mathbf{A}(t, x)$, is also differentiable.

For a given spatial dimension $d$ and speed of light $c$, this linear map (over $\mathbb{R}$) extracts the vector potential component $\mathbf{A}$ from a distributional electromagnetic 4-potential $A$. Given a distributional 4-potential $A$, the resulting vector potential is a distribution defined over space-time ($\text{Time} \times \text{Space } d$) with values in the $d$-dimensional Euclidean space $\mathbb{R}^d$.