is there a definition of Turing computability for multivariate functions?

Question

Suppose we have a function $f:\omega^{n}\rightarrow\omega^{m}$ with $n,m\in\omega$ with $n,m\geq 1$ for which there exists a Turing machine that on input $(k_{1},...,k_{n})$ produces $f(k_{1},...,k_{n})$ as output. My question is: Is there any definition saying that this kind of function are also Turing computable? I know that the case when $m=1$ is the most found in computability books, but my question is what happens in the general case?