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$f: \mathbb R \to \mathbb R^2$ is called a vector-valued function of a real variable. But what is $g:\mathbb R^2 \to \mathbb R^2$? Is it a vector-valued function of a real vector variable? Is vector variable the correct name?

And also, is $h:\mathbb C^2 \to \mathbb C^2$ a complex vector-valued function of a complex vector variable?

JDoeDoe
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    Point of pedantry: $\Bbb R$ can also easily be thought of as a vector space. – Arthur Aug 08 '21 at 09:31
  • Probably more common to call these multivariable functions or functions of several real/complex variables – a1402 Aug 08 '21 at 09:32
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    You could also say $g$ is a real vector-valued function of two real variables, and $h$ is a complex vector-valued function of two complex variables. – sunspots Aug 08 '21 at 09:42
  • Most call functions of the form $g:\mathbb{R}^n\to\mathbb{R}^n$ "vector fields". – K.defaoite Aug 08 '21 at 14:18
  • For a vector variable reference, see page 38 in Advanced Calculus by Loomis and Sternberg: https://people.math.harvard.edu/~shlomo/docs/Advanced_Calculus.pdf. So, there are a variety of ways to describe these functions. – sunspots Aug 09 '21 at 02:15

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Yes. $g$ is a vector-valued function. Interesting the domain and the range of $g$ are both vectors. So, it takes and gives vectors. The inputs and the outputs are both vectors.

The phrase "variable vector" is also not uncommon in literature as can be seen here: variable vector

Regarding your last question, there seems nothing wrong. A more appropriate phrase would be "vector-valued rather than complex vector-valued".

F. A. Mala
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