Images of sets under explicit mappings of R^n

$u = 2 x, v = 3 y$;

$u = a x+a_{0}, v = b y+b_{0}$.

$u = y, v = x y$;

$u = x \cos y, v = x \sin y$.

Find the image of the circle $x^{2}+y^{2} = 2 x$ under the mapping

prove that under this mapping the image of every line and every circle is either a circle or a line;

find the image of the circle $x^{2}+y^{2} = 4$ under the mapping

of the line $x = a$;

of the line $y = b$;

of the domain $x^{2}+y^{2} < 1, x > 0, y > 0$; under the mapping $u = x^{2}-y^{2}, \quad v = 2 x y$.

of the circles: а) $x^{2}+y^{2} = 1$, б) $x^{2}+y^{2} = 1 / 4$, в) $x^{2}+y^{2} = 4$;

of the curve $y = \left|x\right|, y \neq 0$; under the mapping $u = \frac{x}{2}\left(1+\frac{1}{x^{2}+y^{2}}\right), \quad v = \frac{y}{2}\left(1-\frac{1}{x^{2}+y^{2}}\right)$.

of the line $x = a$;

of the segment $x = a,\left|y\right| \leqslant \pi$;

of the line $y = b,\left|b\right| < \pi / 2$;

of the line $y = a x+b, a \neq 0$; under the mapping $u = e^{x} \cos y, \quad v = e^{x} \sin y$.

of the line $x = a$;

of the line $y = b$;

of the half-strip $0 < x < \pi , y > 0 ;$

of the strip $0 < x < \pi$; under the mapping $u = \cos x \operatorname {ch} y, \quad v = \sin x \operatorname {sh} y$.

$u = \sin x, v = \cos 2 x$

$u = a x+a_{0}, v = b x+b_{0}, w = c x+c_{0}$.

$u = y+2, v = 3 x+4 y+5, w = 6 x+7 y+8$;

$u = \cos x \cos y, v = \cos x \sin y, w = \sin x$;

$u = (2+\cos y) \cos x, v = (2+\cos y) \sin x, w = \sin y$.

Show that there are exactly 4 points $\mathbf{p} \in K$ such that

Find these points.

Determine the set of all $\mathbf{q} \in K$ such that

Show that one of the points $\mathbf{p}$ found in part (a) corresponds to a local maximum of $f_{1}$, one corresponds to a local minimum, and that the other two are neither (they are so-called "saddle points").

Which of the points $\mathbf{q}$ found in part $(b)$ correspond to maxima or minima?

Let $\lambda$ be an irrational real number, and define $\mathbf{g}(t) = \mathbf{f}(t, \lambda t)$. Prove that $\mathbf{g}$ is a 1-1 mapping of $\mathbb {R}^{1}$ onto a dense subset of $K$. Prove that