Domains of Functions of Several Variables

$u = \sqrt{x+y}+\sqrt{x-y}$;

$u = \sqrt{1-x^{2}-y^{2}}$;

$u = \frac{1}{\sqrt{x^{2}+y^{2}-1}}$;

$u = \ln \left(x^{2}+y^{2}-1\right)$;

$u = \ln \left(y^{2}-4 x+8\right)$;

$u = \ln \left(x^{2}+4 y^{2}-2 x-3\right)$;

$u = \frac{1}{\sqrt{x^{2}-16\left(y^{2}+1\right)}}$;

$u = \ln \frac{x^{2}+y^{2}-x}{2 x-x^{2}-y^{2}}$;

$u = \sqrt{1-\left|x\right|-\left|y\right|}$;

$u = \frac{1}{5 x^{2}-4 x y+2 y^{2}+2 x-2 y+3}$;

$u = \frac{\ln x \ln y}{\sqrt{1-x-y}}$;

$u = \sqrt{4 x^{2}-x^{3}-4 x}$;

$u = \sqrt{\log_{a}\left(2-x^{2}-y^{2}\right)}$;

$u = \sqrt{\left(x^{2}+y^{2}-a\right)\left(2 a-x^{2}-y^{2}\right)}$;

$u = \ln (3 x+y-3)+\frac{\ln (3-x)}{\sqrt{3 x-2 y+6}}$;

$u = \sqrt{x-1}+\sqrt{y-1}+\sqrt{x+y-3}+\sqrt{42-6 x-7 y}$;

$u = \sqrt{y \sin x}$;

$u = \ln x-\ln \sin y$;

$u = \ln \sin \pi \left(x^{2}+y^{2}\right)$;

$u = \sqrt{x} \ln \operatorname {tg}\left(\frac{\pi y}{x}\right)$;

$u = \arccos y$;

$u = \arccos (x+y)$;

$u = \arcsin \frac{x}{x+y}$;

$u = x^{y}$.

$u = \frac{1}{x^{2}+y^{2}-1}$;

$u = \sqrt{x \sin y}$

$u = \ln \left(1-2 x-x^{2}-y^{2}\right)+\ln \left(1+2 x-x^{2}-y^{2}\right)$;

$u = \arcsin (y / x)$;

$u = \sqrt{x y}+\arcsin x$;

$u = \arccos \left(x / y^{2}\right)$;

$u = \arccos \left(2 y+2 y x^{2}-1\right)$.

$u = \ln (1-x-y-z)$

$u = \sqrt{1-\left|x\right|-\left|y\right|-\left|z\right|}$

$u = \frac{1}{\sqrt{x}}+\frac{1}{\sqrt{z}}$

$u = \frac{1}{\sqrt{1-x^{2}-y^{2}-z^{2}}}$;

$u = \sqrt{16-x^{2}-z^{2}}$;

$u = \sqrt{\left(9-y^{2}-z^{2}\right)\left(y^{2}+z^{2}-4\right)}$;

$u = \ln \left(36-36 x^{2}-9 y^{2}-4 z^{2}\right)$;

$u = \frac{1}{\sqrt{z-x^{2}-y^{2}}}$;

$u = \ln \left(2 z^{2}-6 x^{2}-3 y^{2}-6\right)$;

$u = \sqrt{x^{2}+2 y^{2}+2 y z+z^{2}-2 y+1}$;

$u = \sqrt{16-x^{2}-y^{2}-z^{2}} \ln \left(x^{2}+y^{2}+z^{2}-4\right)$;

$u = \sqrt{2\left(x^{2}+y^{2}+z^{2}\right)-\left(x^{2}+y^{2}+z^{2}\right)^{2}-1}$;

$u = \frac{\ln x+\ln z}{\sqrt{y-1}}+\ln (5-x-y-z)$;

$u = \sqrt{x}+\sqrt{y}+\sqrt{z}+\sqrt{1-x-y}+\sqrt{3 x+y-3 z}$;

$u = \frac{\ln \left(z^{2}-x^{2}-y^{2}\right)}{\sqrt{1-x^{2}-y^{2}-z^{2}}}$;

$u = \frac{\ln \left(x^{2}+y^{2}+z^{2}-4\right)}{\sqrt{4-x^{2}-y^{2}}}$;

$u = \frac{\ln \left(4 z-z^{2}-x^{2}-y^{2}\right)}{\sqrt{4-z-x^{2}-y^{2}}}$;

$u = \sqrt{z-x y}+\sqrt{1-z-\sqrt{x^{2}+y^{2}}}$;

$u = \arccos \left(2-x^{2}-y^{2}-z^{2}\right)$;

$u = \arcsin \frac{\sqrt{x^{2}+y^{2}}}{z-1}$;

$u = \arccos \left(x^{2}+y^{2}+z^{2}-3\right)+\arccos \sqrt{x^{2}+y^{2}-3}$.

$u = \frac{1}{x^{2}+y^{2}+z^{2}}$;

$u = \sqrt{x}+\sqrt{y}+\sqrt{z}$;

$u = \ln (x y z)$;

$u = \sqrt{z^{2}-x^{2}-y^{2}}$;

$y = \sqrt{z^{2}-x^{2}-y^{2}-1}$;

$u = \ln \left(z^{2}-x^{2}-y^{2}+1\right)$;

$u = \ln (y \ln (z-y))$;

$u = \arccos \frac{x^{2}+y^{2}}{z^{2}}$;

$u = (x y)^{z}$;

$u = z^{x y}$.

$u = 1 /\left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}-1\right)$;

$u = \sqrt{1-x_{1}^{2}}+\sqrt{4-x_{2}^{2}}+\sqrt{9-x_{3}^{2}}+\sqrt{16-x_{4}^{2}}$;

$u = \sqrt{7-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}-x_{4}^{2}-2 x_{3}-2 x_{4}}$;

$u = \ln \left(144\left(1-x_{1}^{2}\right)-36 x_{2}^{2}-16 x_{3}^{2}-9 x_{4}^{2}\right)$.

$u = \sum_{i = 1}^{n} \sqrt{1-\left|x_{i}\right|}$

$u = \sqrt{1-\sum_{i = 1}^{n} x_{i}}+\sum_{i = 1}^{n} \sqrt{x_{i}}$

$u = \ln \left(1-\sum_{i = 1}^{n}\left(x_{i}-i\right)^{2}\right)$;

$u = \ln \left(\sum_{i = 1}^{n} x_{i}^{2}+\sum_{i < j = 1}^{n} x_{i} x_{j}\right)$

$u = \sqrt{\log_{a}\left(\sum_{i = 1}^{n} x_{i}^{2}\right)}$;

$u = \sum_{i = 1}^{n} \arcsin \left(x_{i}-i\right)$.