Notation for the domain of $f(\mathbf x,\mathbf y)$?

Question

I have two vectors, $\mathbf x\in \mathbb R^n$ and $\mathbf y \in \mathbb R^m$ and the real-valued function $$ f(\mathbf x,\mathbf y)=f(x_1,\dots, x_n, y_1, \dots y_m) $$ Is the domain $\mathbb R^n\times \mathbb R^m=\mathbb R^{n+m}$ or $\mathbb R^n\times \mathbb R^m=\mathbb R^{n m}$?

Answer 1

The set $\mathbb R^N$ consists of $N$-tuples of real numbers. That is, an element in $\mathbb R^N$ is built from $N$ real numbers.

Clearly, your function has $N = n+m$ and not $nm$ real numbers as input, hence it cannot be $\mathbb R^{nm}$, but it is $$\mathbb R^{n}\times \mathbb R^{m} = \mathbb R ^{n+m}.$$

Answer 2

The domain is $\mathbb{R}^n \times \mathbb{R}^m:$ the easiest way to this about this is that $f$ depends on $m + n$ variables.