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'easy calculation of the area of an ellipse'
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| Title of object: |
easy calculation of the area of an ellipse |
| Canonical Name: |
EasyCalculationOfTheAreaOfAnEllipse |
| Type: |
Definition |
| Created on: |
2006-03-06 16:31:06 |
| Modified on: |
2006-03-15 07:38:02 |
| Classification: |
msc:53A04 |
Revision comment (for changes between this and next version):
| edited last sentence (grammar, spelling, punctuation) |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here |
Content:
Consider the unit circle $\left \{ \right (x,y) \in \mathbb{R}^2 : x^2+y^2\le 1\}$. It's a well known fact that the area of this set is $\pi$.
Now consider the following linear transformation $(x,y)\to(u,v)=(ax,by)$.
The determinant of the transformation is $ab$ and the transformed circle is:
$\left \{ \right (u,v) \in \mathbb{R}^2 : \left (\frac{u}{a} \right )^2 + \left (\frac{v}{b} \right )^2 \le 1\}$ an ellipse of axis $(a,b)$.
Now since the Jacobian of the transformation is constant, the \PMlinkname{change of variables in integral theorem}{ChangeOfVariablesInIntegralOnMathbbRn} allows us to say the area of the transformed set is $ab$ time the area of the original set.
Thus the area of an ellipse if $\pi a b$ |
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