PhyslibAlpha.ClassicalFieldTheory.Local.TotalDerivative

Given a jet order $k \in \mathbb{N}$, a local jet-dependent function $F: J^k(\text{Space } d, \mathbb{R}^m) \to \mathbb{R}$, and a field $f: \text{Space } d \to \mathbb{R}^m$, this definition returns the function from $\text{Space } d$ to $\mathbb{R}$ that maps each point $x$ to $F(j^k f(x))$, where $j^k f(x)$ is the $k$-jet of the field $f$ at $x$.

Given a jet order $k \in \mathbb{N}$, a coordinate index $i \in \{0, \dots, d-1\}$, a local jet-dependent function $F: J^k(\text{Space } d, \mathbb{R}^m) \to \mathbb{R}$, and a field $f: \text{Space } d \to \mathbb{R}^m$, the total derivative is the function from $\text{Space } d$ to $\mathbb{R}$ defined by the partial derivative of the composition $F \circ j^k f$ with respect to the $i$-th coordinate. Specifically, for any point $x \in \text{Space } d$, it evaluates to $$\frac{d}{dx_i} \left( F(j^k f(x)) \right) = \partial_i \left( \text{evalOnJet } k \, F \, f \right)(x)$$ where $j^k f(x)$ is the $k$-jet of the field $f$ at $x$, and $\partial_i$ denotes the spatial derivative in the $i$-th standard basis direction.

Given a spatial dimension $d$, a field dimension $m$, a jet order $k \in \mathbb{N}$, and a multi-index $I \in \mathbb{N}^d$, let $F: J^k(\mathbb{R}^d, \mathbb{R}^m) \to \mathbb{R}$ be a local jet-dependent function and $f: \mathbb{R}^d \to \mathbb{R}^m$ be a field. The iterated total derivative $D^I F$ (evaluated on the field $f$) is the function mapping each point $x \in \mathbb{R}^d$ to the iterated partial derivative of the evaluation of $F$ along the $k$-jet of $f$: $$(D^I F[f])(x) = \frac{\partial^{|I|}}{\partial x^I} \left( F(j^k f(x)) \right) = \frac{\partial^{|I|}}{\partial x_0^{I_0} \partial x_1^{I_1} \cdots \partial x_{d-1}^{I_{d-1}}} \left( F(j^k f(x)) \right)$$ where $j^k f(x)$ denotes the $k$-jet of the field $f$ at the point $x$.

Let $k \in \mathbb{N}$ be a jet order, $F: J^k(\text{Space } d, \mathbb{R}^m) \to \mathbb{R}$ be a local jet-dependent function, and $f: \text{Space } d \to \mathbb{R}^m$ be a field. For any point $x \in \text{Space } d$, the evaluation of $F$ along the $k$-jet of $f$ at $x$ is equal to $F(j^k f(x))$, where $j^k f(x)$ is the $k$-jet of the field $f$ at $x$.

For a jet order $k \in \mathbb{N}$, a coordinate index $i \in \{0, \dots, d-1\}$, a local jet-dependent function $F: J^k(\mathbb{R}^d, \mathbb{R}^m) \to \mathbb{R}$, and a field $f: \mathbb{R}^d \to \mathbb{R}^m$, the total derivative of $F$ in the $i$-th coordinate direction (evaluated along $f$) is equal to the spatial derivative $\partial_i$ of the function that maps each point $x$ to the evaluation of $F$ at the $k$-jet of $f$ at $x$. Mathematically, this is expressed as: $$\text{totalDerivative } k \, i \, F \, f = \partial_i (x \mapsto F(j^k f(x)))$$ where $j^k f(x)$ denotes the $k$-jet of the field $f$ at the point $x$, and $\partial_i$ denotes the partial derivative with respect to the $i$-th spatial coordinate.

For a jet order $k \in \mathbb{N}$, a multi-index $I \in \mathbb{N}^d$, a jet-dependent function $F: J^k(\mathbb{R}^d, \mathbb{R}^m) \to \mathbb{R}$, and a field $f: \mathbb{R}^d \to \mathbb{R}^m$, the iterated total derivative $D^I F$ of $F$ evaluated along the field $f$ is equal to the iterated partial derivative of the composition of $F$ with the $k$-jet of $f$: $$(D^I F[f])(x) = \partial^{[I]} \left( F(j^k f(x)) \right)$$ where $\partial^{[I]}$ denotes the iterated partial derivative $\frac{\partial^{|I|}}{\partial x_0^{I_0} \partial x_1^{I_1} \cdots \partial x_{d-1}^{I_{d-1}}}$ and $j^k f(x)$ is the $k$-jet of the field $f$ at the point $x$.

Given a spatial dimension $d$, a field dimension $m$, and a jet order $k \in \mathbb{N}$, let $F: J^k(\mathbb{R}^d, \mathbb{R}^m) \to \mathbb{R}$ be a local jet-dependent function and $f: \mathbb{R}^d \to \mathbb{R}^m$ be a field. The iterated total derivative $D^I F$ with respect to the zero multi-index $I = 0$ (evaluated on the field $f$) is equal to the evaluation of $F$ along the $k$-jet of $f$. That is, $$(D^0 F[f])(x) = F(j^k f(x))$$ where $j^k f(x)$ denotes the $k$-jet of the field $f$ at the point $x$.

For a jet order $k \in \mathbb{N}$, a coordinate index $i \in \{0, \dots, d-1\}$, a local jet-dependent function $F: J^k(\mathbb{R}^d, \mathbb{R}^m) \to \mathbb{R}$, and a field $f: \mathbb{R}^d \to \mathbb{R}^m$, the iterated total derivative $D^I F[f]$ with respect to the multi-index $I = e_i$ (where $e_i$ is the multi-index with 1 at the $i$-th position and 0 elsewhere) is equal to the total derivative of $F$ along $f$ in the direction $i$. That is, $$(D^{e_i} F[f])(x) = \frac{d}{dx_i} \left( F(j^k f(x)) \right)$$ where $j^k f(x)$ is the $k$-jet of the field $f$ at the point $x$.