Physlib.Relativity.Tensors.RealTensor.Derivative

Given a map $f: \mathbb{R}T(d, c_m) \to \mathbb{R}T(d, c_n)$ between spaces of real Lorentz tensors in dimension $d$ with index structures $c_m$ and $c_n$, `realLorentzTensor.mapToBasis f` is the induced map on their coordinate representations. Specifically, it takes a function $\alpha: \text{ComponentIdx}(c_m) \to \mathbb{R}$ representing the scalar components of a tensor, maps it back to an element of the tensor space $\mathbb{R}T(d, c_m)$ using the canonical basis, applies $f$, and then extracts the scalar components of the resulting tensor as a function $\beta: \text{ComponentIdx}(c_n) \to \mathbb{R}$.

Given a function $f: \mathbb{R}T(d, c_m) \to \mathbb{R}T(d, c_n)$ between spaces of real Lorentz tensors of dimension $d$ with index structures $c_m$ and $c_n$, its derivative $\partial f$ is a function that maps a tensor $y \in \mathbb{R}T(d, c_m)$ to a new tensor of higher rank. The resulting tensor space has an index structure formed by concatenating the dual colors of the input indices $c_m$ (representing the indices of the derivative operator $\partial$) with the original output colors $c_n$. Specifically, for any $y \in \mathbb{R}T(d, c_m)$, the component of the tensor $(\partial f)(y)$ corresponding to the combined multi-index $(b_{in}, b_{out})$ is given by the Fréchet derivative of the $b_{out}$-th component of $f$ in the direction of the $b_{in}$-th basis vector of the domain, evaluated at $y$. In coordinate form, this represents the Jacobian matrix of the components of $f$: $$((\partial f)(y))_{b_{in}, b_{out}} = \frac{\partial f_{b_{out}}}{\partial y_{b_{in}}}(y)$$ where $b_{in}$ are the indices associated with the dualized input colors $\tau(c_m)$ and $b_{out}$ are the indices associated with $c_n$.

The symbol $\partial$ is defined as the notation for the derivative operator `realLorentzTensor.derivative`. This operator acts on functions $f: \mathbb{R}T(d, c_m) \to \mathbb{R}T(d, c_n)$ between spaces of real Lorentz tensors, where $d$ is the spacetime dimension and $c_m, c_n$ describe the tensor indices. The derivative $\partial f$ results in a new tensor-valued function whose output rank is increased by the dual of the indices of the domain.

Let $d, n, m$ be natural numbers representing the spacetime dimension and the ranks of the Lorentz tensors. Let $c_m: \{0, \dots, m-1\} \to \text{Color}$ and $c_n: \{0, \dots, n-1\} \to \text{Color}$ be sequences of Lorentz tensor colors. For a function $f: \mathbb{R}T(d, c_m) \to \mathbb{R}T(d, c_n)$ and a tensor $y \in \mathbb{R}T(d, c_m)$, suppose the coordinate representation of $f$ is differentiable at the coordinates of $y$. Then, for any multi-index $b$ belonging to the component index set of the concatenated color sequence $\tau(c_m) \mathbin{+\!+} c_n$ (where $\tau$ denotes the dual color), the $b$-th component of the Lorentz tensor derivative $(\partial f)(y)$ is equal to the Fréchet derivative of the $b_{out}$-th component of the function $f$ in the direction of the $b_{in}$-th basis vector of the domain, evaluated at the coordinates of $y$. Mathematically, if $b$ is identified with the pair $(b_{in}, b_{out})$ via the isomorphism $\text{ComponentIdx}(\tau(c_m) \mathbin{+\!+} c_n) \cong \text{ComponentIdx}(\tau(c_m)) \times \text{ComponentIdx}(c_n)$, the relation is: $$[(\partial f)(y)]_b = \frac{\partial f_{b_{out}}}{\partial y_{b_{in}}}(y)$$ where the right-hand side is the directional derivative of the $b_{out}$-th coordinate of $f$ with respect to the $b_{in}$-th coordinate of the input.