Physlib.Relativity.Tensors.Evaluation

Let $S$ be a tensor species over a field $k$ and $c$ be a sequence of colors representing a collection of vector spaces $\{V_{c_j}\}_{j=0}^n$. A pure tensor $p \in \text{Pure}(S, c)$ is defined as an element of the dependent product $\prod_{j=0}^n V_{c_j}$, representing the tensor $v_0 \otimes v_1 \otimes \dots \otimes v_n$. Given an index $i \in \{0, \dots, n\}$ and a basis index $b \in \{0, \dots, \dim(V_{c_i}) - 1\}$, the function `evalPCoeff` returns the $b$-th coordinate of the $i$-th component vector $v_i$ with respect to the basis of $V_{c_i}$ provided by the tensor species.

Let $S$ be a tensor species over a field $k$ and $c: \{0, \dots, n\} \to C$ be a sequence of colors representing a collection of vector spaces $\{V_{c_j}\}_{j=0}^n$. Let $p \in \text{Pure}(S, c)$ be a pure tensor, representing the product $v_0 \otimes v_1 \otimes \dots \otimes v_n$. Let $\text{evalPCoeff}(i, b, p)$ be the function that returns the $b$-th coordinate of the $i$-th component vector $v_i$ with respect to the basis of $V_{c_i}$ provided by the species. For any index $i \in \{0, \dots, n\}$, basis index $b \in \{0, \dots, \dim(V_{c_i}) - 1\}$, and vector $x \in V_{c_i}$, the $b$-th basis coefficient of the $i$-th component of the pure tensor updated at index $i$ with $x$ is equal to the $b$-th basis coefficient of $x$: $$\text{evalPCoeff}(i, b, \text{update}(p, i, x)) = \text{repr}(x)_b$$ where $\text{repr}(x)_b$ denotes the $b$-th coordinate of $x$ in the basis of $V_{c_i}$.

Let $S$ be a tensor species over a field $k$ and $c: \{0, \dots, n\} \to C$ be a sequence of colors representing a collection of vector spaces $\{V_{c_m}\}_{m=0}^n$. Let $p$ be a pure tensor, defined as an element of the dependent product $\prod_{m=0}^n V_{c_m}$, which represents the simple tensor $v_0 \otimes v_1 \otimes \dots \otimes v_n$. Let $\text{evalPCoeff}(i, b, p)$ be the function that returns the $b$-th coordinate of the $i$-th component vector $v_i$ with respect to the basis of $V_{c_i}$. For any index $i \in \{0, \dots, n\}$ and $j \in \{0, \dots, n-1\}$, let $k = i.\text{succAbove } j$ denote an index in $\{0, \dots, n\}$ distinct from $i$. If the $k$-th component of $p$ is updated with a new vector $x \in V_{c_k}$, the $b$-th basis coefficient of the $i$-th component vector remains unchanged: $$\text{evalPCoeff}(i, b, \text{update } p \, k \, x) = \text{evalPCoeff}(i, b, p)$$

Let $S$ be a tensor species over a field $k$ and $c: \{0, \dots, n\} \to C$ be a sequence of colors. A pure tensor $p$ of type $c$ is a family of vectors $(v_0, v_1, \dots, v_n)$ representing the tensor product $v_0 \otimes v_1 \otimes \dots \otimes v_n$, where each $v_j$ belongs to the vector space $V_{c_j}$ associated with color $c_j$. Given an index $i \in \{0, \dots, n\}$ and a basis index $b$ for the vector space $V_{c_i}$, the function $\text{evalP } i \, b \, p$ returns the tensor of rank $n$ formed by evaluating the $i$-th index of $p$ at the $b$-th basis element. This is defined as: \[ \text{evalP } i \, b \, p = (v_i)_b \cdot (v_0 \otimes \dots \otimes v_{i-1} \otimes v_{i+1} \otimes \dots \otimes v_n) \] where $(v_i)_b$ is the $b$-th coordinate of the vector $v_i$ with respect to the basis of $V_{c_i}$ provided by the tensor species $S$.

Let $S$ be a tensor species over a field $k$ and $c: \{0, \dots, n\} \to C$ be a sequence of colors, where each $c(m)$ corresponds to a vector space $V_{c(m)}$. Let $p$ be a pure tensor (formally an element of the dependent product $\prod_{m=0}^n V_{c(m)}$) representing the simple tensor $v_0 \otimes \dots \otimes v_n$. Let $\text{evalP}(i, b, p)$ be the operation that evaluates the $i$-th index of $p$ at the $b$-th basis element of $V_{c(i)}$, returning a tensor of rank $n$. For any indices $i, j \in \{0, \dots, n\}$, a basis index $b$ for $V_{c(i)}$, and vectors $x, y \in V_{c(j)}$, the evaluation operation is additive with respect to updating the $j$-th component of the pure tensor: $$\text{evalP}(i, b, \text{update}(p, j, x + y)) = \text{evalP}(i, b, \text{update}(p, j, x)) + \text{evalP}(i, b, \text{update}(p, j, y))$$ where $\text{update}(p, j, v)$ denotes the pure tensor $p$ with its $j$-th component replaced by $v$.

Let $S$ be a tensor species over a field $k$ and $c: \{0, \dots, n\} \to C$ be a sequence of colors, where $V_{c_m}$ denotes the vector space associated with color $c_m$. Let $p$ be a pure tensor of type $c$, representing the family of vectors $(v_0, \dots, v_n)$ where $v_m \in V_{c_m}$. For any indices $i, j \in \{0, \dots, n\}$, a basis index $b$ for the vector space $V_{c_i}$, a scalar $r \in k$, and a vector $x \in V_{c_j}$, the evaluation of the $i$-th index at basis index $b$ is homogeneous with respect to scalar multiplication in the $j$-th component: $$\text{evalP}(i, b, \text{update}(p, j, r \cdot x)) = r \cdot \text{evalP}(i, b, \text{update}(p, j, x))$$ where $\text{update}(p, j, x)$ denotes the pure tensor obtained by replacing the $j$-th component of $p$ with $x$, and $\text{evalP}(i, b, p)$ is the rank-$n$ tensor defined by $(v_i)_b \cdot (v_0 \otimes \dots \otimes v_{i-1} \otimes v_{i+1} \otimes \dots \otimes v_n)$, where $(v_i)_b$ is the $b$-th coordinate of $v_i$ with respect to the basis of $V_{c_i}$.

Let $S$ be a tensor species over a field $k$ and $c: \{0, \dots, n\} \to C$ be a sequence of colors. For a fixed index $i \in \{0, \dots, n\}$ and a basis index $b$ for the vector space $V_{c_i}$ (the space associated with color $c_i$), the function `evalPMultilinear i b` is the multilinear map from the product of vector spaces $\prod_{j=0}^n V_{c_j}$ to the space of tensors of rank $n$ with the color sequence $c \circ i.\text{succAbove}$ (where the $i$-th color is omitted). Given a tuple of vectors $(v_0, \dots, v_n)$, the map is defined by: \[ (v_0, \dots, v_n) \mapsto (v_i)_b \cdot (v_0 \otimes \dots \otimes v_{i-1} \otimes v_{i+1} \otimes \dots \otimes v_n) \] where $(v_i)_b$ is the $b$-th coordinate of the vector $v_i$ with respect to the basis of $V_{c_i}$ provided by the tensor species $S$.

Let $S$ be a tensor species over a field $k$ and $c: \{0, \dots, n\} \to C$ be a sequence of colors. For a fixed index $i \in \{0, \dots, n\}$ and a basis index $b$ for the vector space $V_{c_i}$ associated with the color $c_i$, the $k$-linear map $\text{evalT } i \text{ } b$ maps a tensor $t \in \text{Tensor } S c$ to a tensor of rank $n$ with the color sequence $c \circ i.\text{succAbove}$ (the original sequence $c$ with the $i$-th element omitted). This map is defined as the linear extension of the multilinear evaluation map, which acts on a pure tensor as: \[ v_0 \otimes \dots \otimes v_n \mapsto (v_i)_b \cdot (v_0 \otimes \dots \otimes v_{i-1} \otimes v_{i+1} \otimes \dots \otimes v_n) \] where $(v_i)_b$ is the $b$-th coordinate of the vector $v_i$ with respect to the basis of $V_{c_i}$ provided by the tensor species $S$.

Let $S$ be a tensor species over a field $k$ and $c: \{0, \dots, n\} \to C$ be a sequence of colors. For any index $i \in \{0, \dots, n\}$, any basis index $b$ for the vector space $V_{c_i}$ associated with the color $c_i$, and any pure tensor $p$ (representing the tuple of vectors $(v_0, \dots, v_n)$), the $k$-linear evaluation map $\text{evalT } i \, b$ applied to the tensor $\bigotimes_{j=0}^n v_j$ is equal to the direct evaluation of the pure tensor $\text{evalP } i \, b \, p$. That is, \[ \text{evalT } i \, b (p.\text{toTensor}) = \text{evalP } i \, b \, p \] where $p.\text{toTensor}$ is the image of the vectors in the tensor product space, and $\text{evalP}$ is defined as $(v_i)_b \cdot (v_0 \otimes \dots \otimes v_{i-1} \otimes v_{i+1} \otimes \dots \otimes v_n)$.