PhyslibAlpha.ClassicalFieldTheory.Local.JetPointRegularity

Let $g: \text{Space } d \to \mathbb{R}^m$ be a smooth ($C^\infty$) field. For any natural number $k$, the jet-coordinate map $x \mapsto \text{jetCoordinatesAt } k \, g(x)$ is also smooth ($C^\infty$). This map associates each point $x \in \text{Space } d$ with the collection of jet coordinates $(u^a_I)_{|I| \le k, 0 \le a < m}$, where each coordinate is defined as the partial derivative $u^a_I = \partial^I g_a(x)$ for a multi-index $I$ of order at most $k$ and field component $a$.

Let $g: \mathbb{R}^d \to \mathbb{R}^m$ be a field. If $g$ is $C^\infty$ smooth, then for any $k \in \mathbb{N}$, the map that associates each point $x \in \mathbb{R}^d$ with its base coordinate $x$ and its jet coordinates $(u^a_I)_{|I| \le k, 0 \le a < m}$ at $x$, given by $$x \mapsto \left(x, \left( \partial^I g_a(x) \right)_{|I| \le k, 0 \le a < m} \right)$$ is also $C^\infty$ smooth. Here $I$ denotes a multi-index of order $|I| \le k$ and $g_a$ denotes the $a$-th component of the field $g$.

Let $g : \text{Space } d \to \mathbb{R}^m$ be a field and $k \in \mathbb{N}$ be the maximum derivative order. If a point $x \in \text{Space } d$ is not in the topological support of $g$ (i.e., $x \notin \text{supp}(g)$), then the jet coordinates of order $k$ of the field $g$ at $x$ are zero: $$\text{jetCoordinatesAt } k \, g \, x = 0$$ This implies that for all component indices $a \in \{0, \dots, m-1\}$ and all multi-indices $I$ with $|I| \le k$, the partial derivatives $\partial^{[I]} g_a(x)$ vanish.

Let $g : \text{Space } d \to \mathbb{R}^m$ be a field and $k \in \mathbb{N}$ be the jet order. If a point $x \in \text{Space } d$ is not in the topological support of $g$ (denoted $x \notin \text{supp}(g)$), then the $k$-th order jet fiber direction of the field $g$ at $x$ is zero: $$\text{jetDirectionAt } k \, g \, x = 0$$ This implies that all partial derivatives of the components of $g$ up to order $k$ vanish at the point $x$.