I am looking for a reference for some fairly elementary definitions and calculations about "tensor-valued" functions, i.e. functions of the form $A : \mathbb R^d \to \mathbb R^{d^{n\times}}$.
For example we could define the $i$-th partial derivative of $A$ recursively via $$\partial_i A = (\partial_i A^1, \dots, \partial_i A^d) : \mathbb R^d \to \mathbb R^{d^{n\times}},$$ where $A^i : \mathbb R^d \to \mathbb R^{d{(n-1)\times}}, x\mapsto (A(x))_{i,(\cdot),\dots (\cdot)}$ and then let $$\nabla A = (\partial_1 A,\dots, \partial_d A) : \mathbb R^d \to \mathbb R^{d^{(n+1)\times}}.$$
Then can think about taking a product $Av$ with a function $v : \mathbb R^d \to \mathbb R^d$ and which sense $$\nabla(Av) = \nabla A v + A \nabla v$$ holds. Now $\nabla v$ is not vector-valued anymore, so we also need to know how to multiply tensors with matrices, etc (what happens though if we take $\nabla^2$ e.g.?). Similarly, one could ask about a chain rule in this context.
Of course, you can figure these out on your own, but I would like to have a reference that I can cite (since it's not exactly "new")
What reference deals with these questions?
Motivation: My goal is to formally differentiate an Ito SDE $n$-times w.r.t. to the initial condition. Even if you only differentiate once you need to deal with the derivative of the diffusion coefficient $\sigma : \mathbb R^d \to \mathbb R^{d\times m}$, which should already (properly) tensor-valued. When you try to look only at the "partial derivative" $\partial_i X_t :\mathbb R^d\to \mathbb R^d$ of $X_t :\mathbb R^d\to \mathbb R^d$, the SDE satisfied by $\partial_i X_t$ still incorporates tensor-vector products in the coefficients (and by the product rule, for higher dervatives tensor-tensor products).
