Concerning the notation $\chi (U)$ in one of the hypothesis for some properties of curl and divergence

Question

I have the following exercise:

Let $U \subset \mathbb{R}^3$ be open, $X \in \chi (U)$ and $f \in C^{\infty}(U)$, prove the following: $$curl(\nabla f)=0 \\ div(curl(X))=0 \\ curl(f.X)= f.curl(X)+(\nabla f)× X \\ div(f.X)=f.div(X)+\nabla f • X $$ (Here $×$ and $•$ are the vector and the scalar products respectively)

My question is: what is $\chi (U)$ and why do I need $X \in \chi (U)$? I am not familiar with that notation, but even if I ignore that hypothesis I think I still can prove all those statements. Any suggestion is welcome.

Thanks

Edit: I apologize, I had written composition of functions where actually was product of functions, now it makes sense.

Answer 1

Where did you get the exercise? For example if it's from a book, the notation should be explained in the book. From the context, it's probably just the space of smooth vector fields on U.