Physlib.Electromagnetism.Kinematics.ScalarPotential

Let $d$ be the spatial dimension and $c$ be the speed of light. Suppose $A$ is an electromagnetic four-potential defined on $d$-dimensional spacetime. If $A$ is $n$-times continuously differentiable ($C^n$) with respect to the real numbers, then the scalar potential $\phi$ associated with $A$ (viewed as a function of both time and space) is also $n$-times continuously differentiable ($C^n$).

Let $d$ be the spatial dimension and $c$ be the speed of light. Suppose $A$ is an electromagnetic four-potential defined on $d$-dimensional spacetime. If $A$ is $n$-times continuously differentiable ($C^n$), then for any fixed time $t$, the scalar potential $\phi$ associated with $A$, viewed as a function of space $x \mapsto \phi(t, x)$, is also $n$-times continuously differentiable ($C^n$).

Let $d$ be the spatial dimension and $c$ be the speed of light. Suppose $A$ is an electromagnetic four-potential defined on $d$-dimensional spacetime. If $A$ is infinitely differentiable ($C^\infty$), then for any fixed time $t$ and any natural number $n$, the scalar potential $\phi$ associated with $A$, viewed as a function of space $x \mapsto \phi(t, x)$, is $n$-times continuously differentiable ($C^n$).

Let $d$ be the spatial dimension and $c$ be the speed of light. Suppose $A$ is an electromagnetic four-potential defined on $d$-dimensional spacetime. If $A$ is $n$-times continuously differentiable ($C^n$) with respect to the real numbers, then for any fixed spatial position $x$, the scalar potential $\phi$ associated with $A$ is $n$-times continuously differentiable ($C^n$) as a function of time.

Given a spatial dimension $d$, a speed of light $c$, and an electromagnetic potential $A$ that is differentiable over $\mathbb{R}$, the scalar potential $\phi$ derived from $A$ and $c$ is differentiable over $\mathbb{R}$ as a function of both time and space.

Given a spatial dimension $d$, a speed of light $c$, and an electromagnetic potential $A$ that is differentiable over $\mathbb{R}$, then for any fixed time $t \in \text{Time}$, the scalar potential $\phi(t, \cdot) : \text{Space}_d \to \mathbb{R}$ (derived from $A$ and $c$) is differentiable over $\mathbb{R}$ as a function of space.

Given a spatial dimension $d$, a speed of light $c$, and an electromagnetic potential $A$ that is differentiable over $\mathbb{R}$, the scalar potential $\phi$ associated with $A$ and $c$ is differentiable with respect to time $t$ for any fixed spatial position $x \in \text{Space}_d$.

Given a spatial dimension $d$ and a speed of light $c$, the scalar potential $\phi$ is defined as the real-linear map from the space of electromagnetic potential distributions to the space of real-valued distributions on space-time $\text{Time} \times \text{Space}_d$. For an electromagnetic potential distribution $A$, the scalar potential is obtained by scaling the potential by $c$ and extracting its temporal component, effectively satisfying the relationship $\phi = c A^0$.