Level sets and range of functions of several variables

$u = x^{2}-2 x y+y^{2}+2 x-2 y-3$

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

$u = \ln \left(4 x^{2}+2 y^{2}-4 x y+12 x-12 y+21\right)$;

$u = \log_{y} x+\log_{x} y$;

$u = e^{2 x y}-e^{x y}+2$;

$u = 3 \sin (y / x)+8 \sin^{2}(y / 2 x)$;

$u = 3 \sin (x-y)+6 \sin (x+y)+4 \cos (x-y)+8(x+y)$;

$u = \arccos \frac{1+x^{2} y^{2}}{2 x y}$.

$u = x-2 y-3, E = \left\{ (x ; y): x+y = 1, x \geqslant 0, y \geqslant 0\right\}$;

$u = x^{2}-x y+y^{2}, E = \left\{ (x ; y):\left|x\right|+\left|y\right| = 1\right\}$;

$u = x^{2}+y^{2}-12 x+16 y+25, E = \left\{ (x ; y): x^{2}+y^{2} = 25\right\}$;

$u = \ln \left(2 x^{2}+3 y^{2}\right), E = \left\{ (x ; y): x+y = 2, x \geqslant 0, y \geqslant 0\right\}$;

$u = \sqrt[4]{x^{4}+y^{4}}, E = \left\{ (x ; y): x+y = 2\right\}$.

$u = x^{2}+2 y^{2}+5 z^{2}+2 x y+4 y z-20 z$

$u = x^{2}+y^{2}+3 z^{2}+4 x y+2 x z+2 y z$

$u = \sqrt{x^{2}+y^{2}+2 x+5}+\sqrt{2 x^{2}+z^{2}+4 x+11}$;

$u = \sqrt{4 x^{2}+y^{2}+z^{2}+2 x y-4 z+5}$

$u = 3 \ln (|x|+|y|+|z|)-\ln |x y z|$.

$u = x^{2}+y^{2}+z^{2}, E = \left\{ (x ; y ; z): x+y+z = 6\right\}$;

$u = \frac{1}{x^{2}+y^{2}+z^{2}-2 x+3}, E = \left\{ (x ; y ; z): x+y-z = 1\right\}$;

$u = \sqrt{x^{2}+2 y^{2}+3 z^{2}+2 y z}, E = \left\{ (x ; y ; z): x^{2}+y^{2}+z^{2} = 100\right\}$;

$u = \operatorname {arctg} \sqrt{x y}-(\ln (1+x z)) / 2, E = \left\{ (x ; y ; z): x = y = z\right\}$.

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

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

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

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

$u = y-x$;

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

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

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

$u = \sqrt{36-4 x^{2}-9 y^{2}}$;

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

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

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

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

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

$u = x^{y}$;

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

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

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

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

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

$u = \sqrt{1-2\left|x\right|-\left|y\right|}$;

$u = \left|x\right|+\left|y\right|-\left|x+y\right|$;

$u = \min (x, y)$;

$u = \max (\left|x\right|,\left|y\right|)$;

$y = \min \left(x^{2}, y\right)$;

$u = \sqrt{\operatorname {sign}(\sin x \sin y)}$.

$u = x+2 y+3 z$;

$u = e^{x+2 y+3 z}$;

$u = x^{2}+y^{2}+4 z^{2}$;

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

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

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

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

$u = (x-y)^{2}+z^{2}$;

$u = 1 / x+1 / z$;

$u = z /(x+y+z-1)$;

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

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

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

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

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

$u = \ln (1-\left|x\right|-\left|y\right|-\left|z\right|)$;

$u = \sqrt{\operatorname {sign} \sin \left(x^{2}+y^{2}+z^{2}\right)}$;

$u = \arcsin \sqrt{\left(x^{2}+y^{2}\right) / z^{2}}$.

$u = \frac{x_{1}}{n}$;

$u = \sqrt{n^{2}-\sum_{i = 1}^{n} x_{i}^{2}}$;

$u = x_{1} / \sum_{i = 1}^{n} x_{i}^{2}$;

$u = \left(\left(x_{1}-1\right)^{2}+\sum_{i = 2}^{n} x_{i}^{2}\right) /\left(\left(x_{1}+1\right)^{2}+\sum_{i = 2}^{n} x_{i}^{2}\right)$.

$f = x^{2}+y^{2}, c > 0$;

$f = \ln \left(x^{2}+y^{2}\right)$.