PhyslibAlpha.ClassicalFieldTheory.Local.FirstVariation.Support

Let $f \colon \text{Space } d \to \mathbb{R}^m$ be a field and $\eta$ be an admissible variation. Then the varied field $\text{variedField}(f, \eta, s)$ evaluated at the parameter $s = 0$ is equal to the original field $f$.

Let $f : \text{Space } d \to \mathbb{R}^m$ be a field and $\eta : \text{Space } d \to \mathbb{R}^m$ be an admissible variation. For any real parameters $s, t \in \mathbb{R}$, applying a variation with parameter $t$ to a field that has already been varied by $s$ is equivalent to applying a single variation with parameter $s + t$ to the original field. That is: $$(f + s\eta) + t\eta = f + (s + t)\eta$$ In terms of the varied field function, this is expressed as: $$\text{variedField}(\text{variedField}(f, \eta, s), \eta, t) = \text{variedField}(f, \eta, s + t)$$

Let $g: \text{Space } d \to \mathbb{R}$ be a function. If $g$ is infinitely differentiable ($C^\infty$), then for any index $i \in \{0, \dots, d-1\}$, its partial derivative $\partial_i g$ (the spatial derivative in the direction of the $i$-th standard basis vector) is also infinitely differentiable ($C^\infty$).

Let $g: \text{Space } d \to \mathbb{R}$ be a test function. For any basis index $i \in \{0, \dots, d-1\}$, the spatial derivative $\partial_i g$ is also a test function.

Let $g: \text{Space } d \to \mathbb{R}$ be an infinitely differentiable function (of class $C^\infty$). For any finite sequence of indices $L = [i_1, i_2, \dots, i_n]$ where each $i_j \in \{0, \dots, d-1\}$, the iterated partial derivative $$\partial_{i_1} \partial_{i_2} \dots \partial_{i_n} g$$ is also infinitely differentiable.

Let $g: \text{Space } d \to \mathbb{R}$ be a test function. For any finite list of coordinate indices $L = [i_1, i_2, \dots, i_k]$ where each $i_j \in \{0, \dots, d-1\}$, the iterated partial derivative $\partial_{i_1} \partial_{i_2} \dots \partial_{i_k} g$ is also a test function.

For any smooth function $g : \text{Space } d \to \mathbb{R}$ (where $g \in C^\infty$), any coordinate index $i \in \{0, \dots, d-1\}$, and any list of indices $L = [j_1, j_2, \dots, j_n]$, the partial derivative $\partial_i$ commutes with the iterated partial derivatives defined by $L$. Specifically: $$\partial_{j_1} \partial_{j_2} \cdots \partial_{j_n} (\partial_i g) = \partial_i (\partial_{j_1} \partial_{j_2} \cdots \partial_{j_n} g)$$ where $\partial_\mu$ denotes the spatial derivative in the direction of the $\mu$-th standard basis vector.

Let $\eta: \text{Space } d \to \mathbb{R}^m$ be an admissible variation. For any multi-index $I$ of dimension $d$ with order $|I| \le k$ and any component index $a \in \{0, \dots, m-1\}$, the function mapping a point $x \in \text{Space } d$ to the iterated partial derivative $\partial^{[I]} \eta_a(x)$ is a test function.

Let $\mathcal{L}$ be a Lagrangian of order $k$ for fields mapping $\mathbb{R}^d$ to $\mathbb{R}^m$. Let $f: \mathbb{R}^d \to \mathbb{R}^m$ be a field configuration and $\eta: \mathbb{R}^d \to \mathbb{R}^m$ be an admissible variation. For any multi-index $I$ with $|I| \le k$ and any component index $a \in \{0, \dots, m-1\}$, if the mapping $$x \mapsto \frac{\partial \mathcal{L}}{\partial u^a_I}(j^k f(x))$$ is continuous, then the first-variation density term $$x \mapsto \frac{\partial \mathcal{L}}{\partial u^a_I}(j^k f(x)) \cdot \partial^I \eta^a(x)$$ is integrable over $\mathbb{R}^d$. Here, $j^k f(x)$ denotes the $k$-jet of the field $f$ at point $x$, and $\partial^I \eta^a(x)$ is the iterated partial derivative of the $a$-th component of the variation $\eta$.

Let $d, m, k \in \mathbb{N}$. For a smooth ($C^\infty$) field $f: \mathbb{R}^d \to \mathbb{R}^m$ and an admissible variation $\eta: \mathbb{R}^d \to \mathbb{R}^m$, the map $$(s, x) \mapsto (x, j^k(f + s\eta)(x))$$ from $\mathbb{R} \times \mathbb{R}^d$ to the space of base-jet coordinates $\mathbb{R}^d \times \text{JetCoordinates}(d, m, k)$ is continuous. Here $s \in \mathbb{R}$ is the variation parameter, $x \in \mathbb{R}^d$ is the spatial point, and $j^k(f + s\eta)(x)$ denotes the jet coordinates $(\partial^{[I]} (f_a + s\eta_a)(x))_{|I| \le k, 0 \le a < m}$ of the varied field at $x$.