15-00-SqueezingmathbbRn.tex

\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{SqueezingmathbbRn}
\pmcreated{2013-03-22 17:10:02}
\pmmodified{2013-03-22 17:10:02}
\pmowner{pahio}{2872}
\pmmodifier{pahio}{2872}
\pmtitle{squeezing $\mathbb{R}^n$}
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\pmauthor{pahio}{2872}
\pmtype{Topic}
\pmcomment{trigger rebuild}
\pmclassification{msc}{15-00}
\pmclassification{msc}{15A04}
\pmrelated{circle}
\pmrelated{Ellipse2}
\pmrelated{Circle}
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\pmrelated{VolumeOfEllipsoid}

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\newcommand{\sR}[0]{\mathbb{R}}
\newcommand{\sC}[0]{\mathbb{C}}
\newcommand{\sN}[0]{\mathbb{N}}
\newcommand{\sZ}[0]{\mathbb{Z}}

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\newtheorem{thm}{Theorem}
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\begin{document}
Squeezing the vector space $\mathbb{R}^n$ in the direction of one coordinate axis, \PMlinkname{i.e.}{Ie} multiplying a certain component $x_i$ of all vectors by a non-zero real number $k$, is a linear transformation of $\mathbb{R}^n$.\\

A concrete example of such squeezing and its results is obtained if we squeeze in $\mathbb{R}^2$, i.e. in the Euclidean plane formed by all pairs\, $(x,\,y)$\, of real numbers, every $y$-coordinate by a positive number\, 
$k = \frac{b}{a}$\, where\, $a > b > 0$.\, We may look how this procedure \PMlinkescapetext{acts} on the circle
\begin{align}
x^2+y^2 = a^2.
\end{align}
Since all ordinates of this equation are shrinked by the factor $\displaystyle \frac{b}{a}$ which is less than 1, we must must multiply the new $y$ in equation (1) by the inverse number $\displaystyle\frac{a}{b}$ in \PMlinkescapetext{order} to keep the equation satisfied; then the new $y$ no longer denotes the ordinate of the circle, but rather the ordinate of the squeezed circle. Thus, the equation of the squeezed curve is
$$x^2+\left(\frac{a}{b}\!\cdot\!y\right)^2 = a^2.$$
Simplifying we first obtain
$$x^2+\frac{a^2y^2}{b^2} = a^2,$$
and dividing all \PMlinkescapetext{terms} by $a^2$ yields
\begin{align}
   \frac{x^2}{a^2}+\frac{y^2}{b^2} = 1.
\end{align}
So the resulting curve is an ellipse with semiaxes $a$ and $b$.\\

In the picture below, the circle\, $x^2\!+\!y^2=a^2$\! is drawn in red and the ellipse\, $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$\, in blue.  The angle $t$ is the eccentric anomaly at the point $P$ of the ellipse, which has the \PMlinkname{parametric presentation}{Parameter}\, $x = a\cos{t}$,\, $y = b\sin{t}$.

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\end{center}

Squeezing $\mathbb{R}^3$ in the directions of the $y$-axis and $z$-axis one can get from the sphere
                     $$x^2\!+\!y^2\!+\!z^2 = a^2$$
the ellipsoid
               $$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2} = 1.$$


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