Physlib.SpaceAndTime.Space.DistConst

Let $M$ be a real normed space and $d$ be a natural number representing the dimension of the space. For any fixed vector $m \in M$, the constant distribution $\text{distConst}(d, m)$ is the $M$-valued tempered distribution on $\text{Space } d$ defined by its action on a Schwartz test function $\phi \in \mathcal{S}(\text{Space } d, \mathbb{R})$ as the integral: $$\int_{\text{Space } d} \phi(x) \cdot m \, dx$$ where $dx$ denotes the volume measure on $\text{Space } d$.

Let $M$ be a real normed space and $d$ be a natural number. For any fixed vector $m \in M$, let $\text{distConst}(d, m)$ be the $M$-valued tempered distribution on $\text{Space } d$ that maps a test function $\phi$ to the integral $\int \phi(x) \cdot m \, dx$. Then, for any index $\mu \in \{0, \dots, d-1\}$, the partial derivative of this constant distribution in the direction of the $\mu$-th basis vector, denoted by $\text{distDeriv } \mu$, is zero.

For any dimension $d \in \mathbb{N}$ and any real number $m \in \mathbb{R}$, let $\text{distConst}(d, m)$ be the constant distribution on $\text{Space } d$ that maps a Schwartz test function $\phi$ to the integral $\int_{\text{Space } d} \phi(x) \cdot m \, dx$. Then, the distributional gradient of this constant distribution is zero, i.e., $\nabla (\text{distConst}(d, m)) = 0$.

For any dimension $d \in \mathbb{N}$ and any vector $m$ in the $d$-dimensional Euclidean space $\mathbb{R}^d$, the distributional divergence of the constant distribution associated with $m$ on $\text{Space } d$ is equal to zero.

Let $m \in \mathbb{R}^3$ be a constant vector. Let $\text{distConst}(3, m)$ be the constant distribution on the $3$-dimensional space $\text{Space } 3$ (isomorphic to $\mathbb{R}^3$) that maps a Schwartz test function $\phi \in \mathcal{S}(\text{Space } 3, \mathbb{R})$ to the integral $\int_{\text{Space } 3} \phi(x) \cdot m \, dx$. Then the distributional curl of this constant distribution is the zero distribution: $$\text{distCurl}(\text{distConst}(3, m)) = 0.$$