PhyslibAlpha.ClassicalFieldTheory.Local.JetPointFiber

For an object $V$ of type `JetFiberData d m k`, this instance defines a coercion that allows $V$ to be treated as a function. Specifically, for a multi-index $I \in \mathbb{N}^d$ with total order $|I| \le k$ and a field component index $a \in \{0, \dots, m-1\}$, $V(I, a)$ returns the real-valued coordinate $V^I_a$ associated with the $a$-th field component and the derivative multi-index $I$.

For any two objects $V, W$ of type `JetFiberData d m k`, representing coordinate-level data for a jet of order $k$ with $m$ field components in $d$ dimensions, if their coordinate values $V.\text{coord}(I, a)$ and $W.\text{coord}(I, a)$ are equal for all multi-indices $I$ with total order $|I| \leq k$ and all field component indices $a \in \{0, \dots, m-1\}$, then $V$ and $W$ are equal.

For an element $V$ of the jet fiber data $\text{JetFiberData}(d, m, k)$, the function `value` returns the zero-th order component as a vector in $\mathbb{R}^m$ (modeled as `EuclideanSpace ℝ (Fin m)`). For each component $a \in \{0, \dots, m-1\}$, the $a$-th entry of the resulting vector is given by the coordinate $V(0, a)$, where $0$ is the zero multi-index.

For any jet fiber data $V \in \text{JetFiberData}(d, m, k)$ and any field component index $a \in \{0, \dots, m-1\}$, the $a$-th component of the field value vector $V.\text{value} \in \mathbb{R}^m$ is equal to the coordinate $V.\text{coord}(0, a)$, where $0$ denotes the zero multi-index $(0, \dots, 0)$.

The zero element of the space of jet fiber data $\text{JetFiberData}(d, m, k)$ is defined by setting all its coordinate components to $0$.

The addition of two jet fiber data objects $V, W \in \text{JetFiberData}(d, m, k)$ is defined component-wise. For any derivative index $I$ and field component index $a$, the coordinate of the sum is given by the sum of the individual coordinates: $(V + W)_{I, a} = V_{I, a} + W_{I, a}$.

The definition provides a scalar multiplication of `JetFiberData d m k` by a real number. For a scalar $c \in \mathbb{R}$ and an element $V$ of `JetFiberData d m k`, the product $c \cdot V$ is defined such that its coordinate at derivative index $I$ and field component index $a$ is given by $(c \cdot V)_{I, a} = c \cdot V_{I, a}$.

For any multi-index $I$ of dimension $d$ with total order $|I| \le k$ and any field component index $a \in \{0, \dots, m-1\}$, the coordinate of the zero element $0$ in the space of jet fiber data $\text{JetFiberData}(d, m, k)$ at $(I, a)$ is $0$.

For any jet fiber data $V, W \in \text{JetFiberData}(d, m, k)$, a multi-index $I$ with total order $|I| \le k$, and a field component index $a \in \{0, \dots, m-1\}$, the coordinate of the sum $V + W$ at $(I, a)$ is equal to the sum of their individual coordinates: $$(V + W)_{I, a} = V_{I, a} + W_{I, a}$$

For any real number $c \in \mathbb{R}$, any jet fiber data $V$ (parameterized by base dimension $d$, field dimension $m$, and jet order $k$), any derivative multi-index $I$ with total order $|I| \le k$, and any field component index $a$, the coordinate of the scalar product $c \cdot V$ at $(I, a)$ is equal to $c$ times the coordinate of $V$ at $(I, a)$: $$(c \cdot V)_{I, a} = c \cdot V_{I, a}$$

Given a jet point $J$ and a fiber-direction increment $V$ (of type `JetFiberData`), both characterized by base dimension $d$, field dimension $m$, and jet order $k$, this operation returns a new jet point. The resulting jet point has the same base coordinates as $J$, and its fiber coordinates are the sum of the fiber coordinates of $J$ and the coordinates of $V$. Specifically: $$(J + V).\text{base} = J.\text{base}$$ $$(J + V).\text{fiber}_{I, a} = J.\text{fiber}_{I, a} + V.\text{coord}_{I, a}$$ where $I$ is a derivative multi-index and $a$ is a field component index.

Given a jet point $J$, a fiber-direction increment $V$ (both characterized by base dimension $d$, field dimension $m$, and jet order $k$), and a scalar $s \in \mathbb{R}$, this function returns the jet point representing the affine line in the jet space fiber passing through $J$ in the direction $V$. The resulting point has the same base coordinates as $J$, while its fiber coordinates are shifted by $s \cdot V$: $$(J + s \cdot V).\text{base} = J.\text{base}$$ $$(J + s \cdot V).\text{fiber}_{I, a} = J.\text{fiber}_{I, a} + s \cdot V.\text{coord}_{I, a}$$ where $I$ is a derivative multi-index and $a$ is a field component index.

For any jet point $J$ and fiber-direction increment $V$, both characterized by base dimension $d$, field dimension $m$, and jet order $k$, the base coordinates of the jet point resulting from adding $V$ to $J$ are equal to the base coordinates of $J$. That is, $(J + V)_{\text{base}} = J_{\text{base}}$.

For any jet point $J$ and fiber-direction increment $V$, both characterized by base dimension $d$, field dimension $m$, and jet order $k$, the zero-th order value of the jet point resulting from the addition of $V$ to $J$ is equal to the sum of the individual values of $J$ and $V$. That is, $$(J + V).\text{value} = J.\text{value} + V.\text{value}$$ where the value represents the field components in $\mathbb{R}^m$ corresponding to the zero multi-index.

Let $J$ be a jet point in the local $k$-jet bundle with base dimension $d$ and field dimension $m$. Let $V$ be a fiber data increment of type `JetFiberData`. For any derivative multi-index $I$ with $|I| \le k$ and any field component index $a \in \{0, \dots, m-1\}$, the coordinate of the translated jet point $J + V$ (denoted as `J.addFiber V`) is given by the sum of the respective coordinates: $$(J + V)^a_I = J^a_I + V^a_I$$ where $J^a_I$ denotes the $(I, a)$-th coordinate of the jet point $J$, and $V^a_I$ denotes the $(I, a)$-th coordinate of the fiber increment $V$.

Let $J$ be a jet point in the local $k$-jet bundle with base dimension $d$ and field dimension $m$. For any fiber-direction increment $V$ (of type `JetFiberData`) and any scalar $s \in \mathbb{R}$, the base coordinate of the jet point resulting from the affine line map $J + s \cdot V$ is equal to the base coordinate of the original point $J$: $$(J + s \cdot V).\text{base} = J.\text{base}$$

Let $J$ be a jet point in the local $k$-jet bundle with base dimension $d$ and field dimension $m$. Let $V$ be a fiber-direction increment of type `JetFiberData` and $s \in \mathbb{R}$ be a scalar. For any derivative multi-index $I$ with $|I| \le k$ and any field component index $a \in \{0, \dots, m-1\}$, the $(I, a)$-th coordinate of the jet point resulting from the affine line map $J.lineMap(V, s)$ is given by: $$(J + s \cdot V)^a_I = J^a_I + s \cdot V^a_I$$ where $J^a_I$ denotes the $(I, a)$-th coordinate of the jet point $J$, and $V^a_I$ denotes the $(I, a)$-th coordinate of the fiber increment $V$.

Given a field $g: \text{Space } d \to \mathbb{R}^m$ and a base point $x \in \text{Space } d$, the function returns the fiber data (of type `JetFiberData`) corresponding to the $k$-th order jet of the field $g$ at $x$. The components $(V^a_I)$ of this fiber data are given by the partial derivatives of the field: $$V^a_I = \partial^{[I]} g_a(x)$$ where $a \in \{0, \dots, m-1\}$ is the field component index and $I$ is a multi-index of order $|I| \le k$. This represents the vertical/fiber part of the jet $j^k g(x)$ in the local coordinate representation.

Let $g: \text{Space } d \to \mathbb{R}^m$ be a field and $x \in \text{Space } d$ be a base point. For any multi-index $I$ with total order $|I| \le k$ and any field component index $a \in \{0, \dots, m-1\}$, the $(I, a)$-th coordinate of the $k$-th order jet fiber direction of $g$ at $x$ is equal to the iterated partial derivative of the $a$-th component of $g$ evaluated at $x$: $$(\text{jetDirectionAt } k \, g \, x)^a_I = \partial^{[I]} g_a(x)$$ where $g_a(x)$ denotes the $a$-th component of the field value $g(x)$, and $\partial^{[I]}$ denotes the partial derivative operator $\frac{\partial^{|I|}}{\partial x_0^{I_0} \cdots \partial x_{d-1}^{I_{d-1}}}$.

Let $g: \text{Space } d \to \mathbb{R}^m$ be a field, $x \in \text{Space } d$ be a point, and $k \in \mathbb{N}$ be the jet order. For any component index $a \in \{0, \dots, m-1\}$, the coordinate of the jet fiber direction $\text{jetDirectionAt}(k, g, x)$ corresponding to the zero multi-index $I = 0$ is equal to the value of the $a$-th component of the field at $x$. That is, $$(\text{jetDirectionAt}(k, g, x))_{0, a} = g_a(x)$$ where $g_a(x)$ denotes the $a$-th component of the field $g$ evaluated at $x$.

Let $f, g: \text{Space } d \to \mathbb{R}^m$ be $C^\infty$ smooth fields. For any point $x \in \text{Space } d$ and scalar $s \in \mathbb{R}$, the jet coordinates of order $k$ of the linear combination $f + s \cdot g$ at the point $x$ are equal to the linear combination of the jet coordinates of $f$ and $g$ at $x$. That is, $$\text{jetCoordinatesAt}_k(f + s \cdot g, x) = \text{jetCoordinatesAt}_k(f, x) + s \cdot \text{jetCoordinatesAt}_k(g, x)$$ where $\text{jetCoordinatesAt}_k(f, x)$ denotes the set of partial derivatives $\{ \partial^I f^a(x) \}$ for multi-indices $I$ with $|I| \le k$ and components $a \in \{0, \dots, m-1\}$.

Let $f, g: \text{Space } d \to \mathbb{R}^m$ be $C^\infty$ smooth fields. For any jet order $k \in \mathbb{N}$, base point $x \in \text{Space } d$, and scalar $s \in \mathbb{R}$, the $k$-jet of the field $y \mapsto f(y) + s \cdot g(y)$ at $x$ is equal to the affine line map of the $k$-jet of $f$ at $x$ shifted by the jet fiber direction of $g$ at $x$ scaled by $s$. Mathematically, this is expressed as: $$j^k(f + s \cdot g)(x) = j^k f(x) + s \cdot \text{jetDirectionAt}_k(g, x)$$ where $j^k f(x)$ is the jet point of $f$ at $x$, and $\text{jetDirectionAt}_k(g, x)$ represents the fiber-coordinate data (the partial derivatives) of $g$ at $x$.