PhyslibAlpha.ClassicalFieldTheory.Local.JetPoint

For a base space of dimension $d$, a field with $m$ components, and a maximum derivative order $k$, `JetCoordinates d m k` is the space of real-valued coordinates $u^a_I$. These coordinates are indexed by multi-indices $I = (I_0, \dots, I_{d-1})$ with total order $|I| = \sum I_i \le k$ and component indices $a \in \{0, \dots, m-1\}$. In local coordinate expressions, an element of this type represents the set of values $\{u^a_I\}_{|I| \le k, 0 \le a < m}$ that characterize the fiber of a $k$-jet bundle.

Given a jet point $J$ in the local $k$-jet bundle, a multi-index $I$ with order $|I| \le k$, and a field component index $a \in \{0, \dots, m-1\}$, this function returns the jet coordinate $u^a_I \in \mathbb{R}$. This coordinate represents the value associated with the $I$-th partial derivative of the $a$-th component of a field at the base point of $J$.

Given a jet point $J$ in the local $k$-jet bundle for fields with values in $\mathbb{R}^m$, this function returns the zero-th order value of the field as a vector in $\mathbb{R}^m$. For each component index $a \in \{0, \dots, m-1\}$, the value is the jet coordinate $u^a_0$ corresponding to the zero multi-index $I = (0, \dots, 0)$.

For a jet point $J$ in the local $k$-jet bundle of fields with $m$ components on a $d$-dimensional space, and for any field component index $a \in \{0, \dots, m-1\}$, the $a$-th component of the zero-th order value of $J$ is equal to the jet coordinate $u^a_0$ corresponding to the zero multi-index $I = 0$: $$(J.\text{value})_a = J.\text{coord}(0, a)$$ where $0$ denotes the multi-index $(0, \dots, 0)$.

Let $J$ and $K$ be jet points in the local $k$-jet bundle of fields with $m$ components on a $d$-dimensional space. If $J$ and $K$ have the same base point ($J.\text{base} = K.\text{base}$) and all their jet coordinates are equal, such that $J.\text{coord}(I, a) = K.\text{coord}(I, a)$ for every multi-index $I$ of order $|I| \le k$ and every field component index $a \in \{0, \dots, m-1\}$, then the jet points are identical, $J = K$.

This function constructs a jet point in the local $k$-jet bundle $J^k(\text{Space } d, \mathbb{R}^m)$. Given a base point $x \in \text{Space } d$ and a set of jet coordinates $u \in \text{JetCoordinates } d m k$ (which represents the field components and their derivatives $\{u^a_I\}_{|I| \le k, 0 \le a < m}$), it returns a jet point $J$ with base point $x$ and fiber coordinates $u$.

For a jet point $J$ of order $k$ associated with a field with $m$ components on a $d$-dimensional base space, this function returns the pair $(x, u)$ consisting of the base point $x \in \mathbb{R}^d$ and the jet coordinates $u = \{u^a_I\}_{|I| \le k, 0 \le a < m}$ representing the values of the field and its derivatives at $x$.

For any base point $x \in \text{Space } d$ and any set of jet coordinates $u \in \text{JetCoordinates } d m k$, the base point of the jet point constructed via `ofBaseCoordinates x u` is equal to $x$.

For any base point $x \in \text{Space } d$ and any set of jet coordinates $u \in \text{JetCoordinates } d, m, k$, the fiber component of the jet point constructed using `ofBaseCoordinates x u` is equal to $u$.

For a base point $x \in \text{Space } d$ and a collection of jet coordinates $u \in \text{JetCoordinates } d m k$, let $J$ be the jet point constructed from $x$ and $u$. For any multi-index $I$ with total order $|I| \le k$ and any field component index $a \in \{0, \dots, m-1\}$, the coordinate $u^a_I$ of the jet point $J$ is equal to the $(I, a)$-th entry of the original coordinate collection $u$.

Let $J$ be a jet point in the local $k$-jet bundle $J^k(\text{Space } d, \mathbb{R}^m)$. The jet point reconstructed from $J$'s base point $x = J.\text{base}$ and its fiber coordinates $u = J.\text{fiber}$ is equal to the original jet point $J$.

For any base point $x \in \text{Space } d$ and jet coordinates $u \in \text{JetCoordinates } d m k$, applying the `toBaseCoordinates` map to the jet point constructed from $x$ and $u$ (via `ofBaseCoordinates`) returns the original pair $(x, u)$.

Given a maximum derivative order $k \in \mathbb{N}$, a field $g: \text{Space } d \to \mathbb{R}^m$, and a base point $x \in \text{Space } d$, the function returns the jet coordinates $(u^a_I)_{|I| \le k, 0 \le a < m}$ at the point $x$. Each coordinate $u^a_I$ is defined as the iterated partial derivative of the $a$-th component of the field $g$ evaluated at $x$: $$u^a_I = \partial^{[I]} g_a(x)$$ where $g_a$ is the $a$-th component of $g$, and $I$ is a multi-index of order $|I| \le k$.

For a given maximum derivative order $k \in \mathbb{N}$, a field $f: \text{Space } d \to \mathbb{R}^m$, and a base point $x \in \text{Space } d$, the function returns the $k$-jet point of $f$ at $x$, often denoted $j^k f(x)$. This jet point is an element of the local jet bundle $J^k(\text{Space } d, \mathbb{R}^m)$ consisting of the base point $x$ and the jet coordinates $u^a_I = \partial^{[I]} f_a(x)$ for all components $0 \le a < m$ and multi-indices $I$ such that $|I| \le k$.

For any maximum derivative order $k \in \mathbb{N}$, any field $f: \text{Space } d \to \mathbb{R}^m$, and any point $x \in \text{Space } d$, the base point component of the $k$-jet of $f$ at $x$ (denoted $j^k f(x)$) is equal to $x$.

For any derivative order $k \in \mathbb{N}$, any field $f: \text{Space } d \to \mathbb{R}^m$, and any base point $x \in \text{Space } d$, the zero-th order value of the $k$-jet of $f$ at $x$ is equal to the value of the field at that point, i.e., $(j^k f(x)).\text{value} = f(x)$.

For any natural number $k$, a field $f: \text{Space } d \to \mathbb{R}^m$, and a base point $x \in \text{Space } d$, let $j^k f(x)$ be the $k$-jet of $f$ at $x$. For any multi-index $I$ with order $|I| \le k$ and any component index $a \in \{0, \dots, m-1\}$, the $(I, a)$-th jet coordinate of $j^k f(x)$ is equal to the iterated partial derivative of the $a$-th component of $f$ at $x$: $$(j^k f(x))^a_I = \partial^{[I]} f_a(x)$$ where $f_a$ denotes the $a$-th component of the field $f$, and $\partial^{[I]}$ is the iterated partial derivative operator associated with the multi-index $I$.

For a maximum derivative order $k \in \mathbb{N}$, a field $f : \text{Space } d \to \mathbb{R}^m$, a base point $x \in \text{Space } d$, and a component index $a \in \{0, \dots, m-1\}$, the jet coordinate associated with the zero multi-index $0$ and the $a$-th component of the $k$-jet of $f$ at $x$ is equal to the $a$-th component of the field value at $x$. That is, $$(j^k f(x))^a_0 = f_a(x)$$ where $j^k f(x)$ denotes the $k$-jet of $f$ at $x$, and $f_a(x)$ denotes the $a$-th component of the field $f$ evaluated at $x$.

For a field $g: \text{Space } d \to \mathbb{R}^m$, a point $x \in \text{Space } d$, a multi-index $I$ with order $|I| \le k$, and a component index $a \in \{0, \dots, m-1\}$, the $(I, a)$-th jet coordinate of $g$ at $x$ is given by the iterated partial derivative of the $a$-th component of $g$ at $x$: $$\text{jetCoordinatesAt}(k, g, x)_{I, a} = \partial^{[I]} g_a(x)$$ where $g_a$ is the $a$-th component of the field $g$, and $\partial^{[I]}$ denotes the iterated partial derivative operator defined by the multi-index $I$.

For a field $g: \text{Space } d \to \mathbb{R}^m$, a base point $x \in \text{Space } d$, and a maximum derivative order $k \in \mathbb{N}$, the jet coordinate associated with the zero multi-index $I = 0$ and the $a$-th component of the field is equal to the $a$-th component of the value of the field at $x$. That is, $$(\text{jetCoordinatesAt } k \ g \ x)^a_0 = g_a(x)$$ where $g_a(x)$ denotes the $a$-th component of the vector $g(x) \in \mathbb{R}^m$.

For any maximum derivative order $k \in \mathbb{N}$, a field $f: \text{Space } d \to \mathbb{R}^m$, and a base point $x \in \text{Space } d$, the $k$-jet of the field $f$ at $x$ (denoted $j^k f(x)$) is equal to the jet point constructed from the base point $x$ and the jet coordinates of $f$ at $x$. Here, the jet coordinates consist of the partial derivatives $(\partial^{[I]} f_a(x))$ for all field components $0 \le a < m$ and multi-indices $I$ such that $|I| \le k$.