While trying to answer my other question I found I never heard about vector-valued differential forms. I've been searching for them in various mathematical physics books, but didn't get too much.
I'm interested in notion and names of two operations with $p$-form valued $n$-forms.
I treat p-form valued n-form as
$ (TM)^n \to \Omega^p(M) $
or
$ (TM)^n \to (TM)^p \to \mathbb R $
I think ususal contraction with a vector field is denoted as $\rfloor$:
$\rfloor : TM \to \left(TM^n \to TM^p \to \mathbb R \right) \to \left(TM^{n-1} \to TM^{p} \to \mathbb R \right) $
How do I denote the similar operation, but when contracting with the resulting $p$-form? :
$\lfloor : TM \to \left(TM^n \to TM^p \to \mathbb R \right) \to \left(TM^n \to TM^{p-1} \to \mathbb R \right) $
Or maybe there is a notation for switching position, that is $p$-form valued $n$-forms is turned into $n$-form valued $p$-forms.
Given an $n$-form and a $p$-form I can produce $p$-form valued $n$-form. How do I denote and name this operation?
$(TM^n \to \mathbb R) \to (TM^p \to \mathbb R) \to \left( TM^n \to TM^p \to \mathbb R \right)$
$(\omega,\nu) \mapsto \left(\left(v^n,u^p\right) \mapsto \omega(v^n) \nu(u^p)\right)$
Maybe these operations are not specific only to this setting and have some universal nature and notation.