Physlib.SpaceAndTime.Space.Derivatives.Iterated

Given a natural number $d$, a multi-index $I = (I_0, I_1, \dots, I_{d-1}) \in \mathbb{N}^d$, and a function $f: \text{Space } d \to M$ mapping into a real topological vector space $M$, the iterated coordinate derivative $\partial^{[I]} f$ is the function obtained by repeatedly applying the partial derivative operator $\partial_\mu$ (defined as `Space.deriv`) for each coordinate index $\mu$ in the canonical list associated with $I$. This corresponds to the standard partial derivative of $f$ of order $|I|$ indexed by the multi-index $I$, denoted as: $$\partial^{[I]} f = \frac{\partial^{|I|} f}{\partial x_0^{I_0} \partial x_1^{I_1} \cdots \partial x_{d-1}^{I_{d-1}}}$$

The notation $\partial^{[I]}$ denotes the iterated coordinate derivative operator indexed by a multi-index $I$. For a function $f: \text{Space } d \to M$, where $M$ is a module over $\mathbb{R}$, $\partial^{[I]} f$ represents the repeated application of the coordinate derivative `Space.deriv` along the directions specified by the multi-index $I$.

Let $M$ be a real topological vector space and $d$ be a natural number. For any function $f: \text{Space } d \to M$, the iterated coordinate derivative with respect to the zero multi-index $0$ is the function itself: $$\partial^{[0]} f = f$$

Let $M$ be a real topological vector space and $f: \text{Space}(d+1) \to M$ be a function. For any multi-index $I = (I_0, \dots, I_d) \in \mathbb{N}^{d+1}$, the iterated coordinate derivative of $f$ indexed by the multi-index incremented at the $0$-th coordinate is given by: $$\partial^{[I + e_0]} f = \partial_0 (\partial^{[I]} f)$$ where $I + e_0$ denotes the multi-index $(I_0 + 1, I_1, \dots, I_d)$, $\partial^{[I]} f$ is the iterated coordinate derivative of $f$ with respect to $I$, and $\partial_0$ is the partial derivative with respect to the $0$-th coordinate.

Let $f: \text{Space } d \to M$ be a function mapping into a real topological vector space $M$, and let $i \in \{0, \dots, d-1\}$ be a coordinate index. Let $e_i$ denote the unit multi-index with $1$ at the $i$-th position and $0$ elsewhere (defined as the increment of the zero multi-index at $i$). Then the iterated coordinate derivative $\partial^{[e_i]} f$ is equal to the spatial derivative in the $i$-th direction $\partial_i f$.

Let $d$ be a natural number and $I \in \mathbb{N}^d$ be a multi-index of dimension $d$. For any two infinitely differentiable (smooth) functions $f, g: \text{Space } d \to \mathbb{R}$, the iterated coordinate derivative of their sum is equal to the sum of their individual iterated coordinate derivatives: $$\partial^{[I]} (f + g) = \partial^{[I]} f + \partial^{[I]} g$$

Let $d$ be a natural number and $I = (I_0, I_1, \dots, I_{d-1}) \in \mathbb{N}^d$ be a multi-index. For any real constant $c \in \mathbb{R}$ and any smooth function $f: \text{Space } d \to \mathbb{R}$ (i.e., $f \in C^\infty$), the iterated coordinate derivative of the scalar multiple $c \cdot f$ satisfies: $$\partial^{[I]} (c \cdot f) = c \cdot \partial^{[I]} f$$ where $\partial^{[I]}$ denotes the partial derivative operator $\frac{\partial^{|I|}}{\partial x_0^{I_0} \partial x_1^{I_1} \cdots \partial x_{d-1}^{I_{d-1}}}$.

Let $d$ be a natural number and $\text{Space } d$ be the $d$-dimensional real inner product space. For any multi-index $I \in \mathbb{N}^d$ and any smooth scalar-valued function $f: \text{Space } d \to \mathbb{R}$ (i.e., $f$ is $C^\infty$), the iterated coordinate derivative $\partial^{[I]} f$ is also a smooth function.

Let $f: \text{Space } d \to \mathbb{R}$ be a function and $i \in \{0, \dots, d-1\}$ be a coordinate index. Let $\partial_i f$ denote the spatial derivative of $f$ in the $i$-th coordinate direction. Then the topological support of the spatial derivative is contained in the topological support of the original function: \[ \operatorname{tsupport}(\partial_i f) \subseteq \operatorname{tsupport}(f) \]

Let $f: \text{Space } d \to \mathbb{R}$ be an infinitely differentiable (smooth) scalar-valued function. For any coordinate index $i \in \{0, \dots, d-1\}$ and any multi-index $I \in \mathbb{N}^d$, the spatial derivative $\partial_i$ commutes with the iterated multi-index derivative $\partial^{[I]}$. That is, \[ \partial_i (\partial^{[I]} f) = \partial^{[I]} (\partial_i f) \] where $\partial_i$ is the partial derivative with respect to the $i$-th coordinate and $\partial^{[I]}$ is the iterated coordinate derivative defined by the multi-index $I$.

Let $d$ be a natural number and $I = (I_0, I_1, \dots, I_{d-1}) \in \mathbb{N}^d$ be a multi-index. For any real-valued function $f: \text{Space } d \to \mathbb{R}$, the topological support of its iterated coordinate derivative $\partial^{[I]} f$, defined as $$\partial^{[I]} f = \frac{\partial^{|I|} f}{\partial x_0^{I_0} \partial x_1^{I_1} \cdots \partial x_{d-1}^{I_{d-1}}},$$ is a subset of the topological support of $f$. That is, $$\text{supp}(\partial^{[I]} f) \subseteq \text{supp}(f).$$

Let $d$ be a natural number and $I = (I_0, I_1, \dots, I_{d-1}) \in \mathbb{N}^d$ be a multi-index. For any real-valued function $f: \text{Space } d \to \mathbb{R}$ and any point $x \in \text{Space } d$, if $x$ is not in the topological support of $f$ (denoted $\text{supp}(f)$), then the iterated coordinate derivative $\partial^{[I]} f$ vanishes at $x$: $$\partial^{[I]} f(x) = 0$$ where $\partial^{[I]} f$ is defined as: $$\partial^{[I]} f = \frac{\partial^{|I|} f}{\partial x_0^{I_0} \partial x_1^{I_1} \cdots \partial x_{d-1}^{I_{d-1}}}$$