# elementary question about multivariable calculus

I have a rather elementary question about multivariable calculus.

In a function $f(x,y)$; are the two values for the variables arbitrary chosen from a given set of numbers or are there any relationships or restrictions beyond that?

What I mean as an example: $f(x,y)=x^2+y$, where $x$ and $y$ are elements of all real numbers. Is there any further restriction to this or any combination rule for $x$ and $y$ that is just given in such a function ?

• "or any combination rule for y and y that is just given in such a function" Did you mean "for x and y"? Also, to answer the question, No - both variables are viewed independently, they can both vary along all real numbers. Apr 13, 2018 at 14:00

No. $x$ and $y$ can be arbitrary real numbers, as long as the expression makes sense. Sometimes the domain may be smaller - just as in the one variable case. The domain for $f(x) = \sqrt{x}$ isn't the whole real line. The domain for $f(x,y) = \sqrt{x^2 + y}$ isn't the whole plane.
For any given function $$f(x)$$, the set acceptable values of $$x$$ on the real line is called the domain $$\mathrm D_f$$ of the funcion $$f$$, i.e., $$\mathrm D_f\subset \mathbb R$$ By 'acceptable', I mean the values of $$x$$ for which the function $$f$$ is defined, consider the example $$f(x)= \sqrt x$$ Clearly, $$f(x)$$ is defined $$\forall x>0$$. Therefore, $$D_f=\{x:x>0,x\in \mathbb R\}$$.
Similarly, for the function $$f(x,y)$$, the permissible values of $$(x,y)$$ make the domain $$D_f\in\mathbb R^2$$.
The values of $$x$$ and $$y$$ are such that they satisfy mathematical combination, which is defined by the function $$f$$.