Volume of N-Dimensional Ellipsoid and Its Extension Article

Volume of n-dimensional ellipsoid and its file format

Team Registration Number: 90014 School Term: Methodist University Teacher's Brand: Mr. Yeung Sik Ming Team Members' Names: Wu Ming Put up, Tsang Person Ho, Lam Cheuk Yan, Mak Sze Long

A report submitted towards the Scientific Panel of the Suspend Lung Math Award, 2010

7th Aug 2010

Subjective

In this conventional paper, we want to locate the mixture of the volume with the n-dimensional ellipsoid. First, we discover the volumes of prints of the 2-dimensional and third-dimensional ellipsoids simply by double and triple integrals. After that, we solve the multiple important having integrand with different exponents e1, e2,

пѓ¦ you пѓ¶, sobre пѓЋ пѓ§ пЂ­, п‚Ґ пѓ·: пѓЁ 2 пѓё

пѓІ пЂЁ пЂ©пЂЁ by пЂ© пЂЁx пЂ©

Sobre a a couple of e1 1 2

two e2

пЂЁx пЂ©

in

2 en

dпЃі and.

dxn пЂ®пЂ

пѓ¬ whereпЂ En a пЂЅ пѓ­пЂЁ x1, пѓ®

пЂЁпЂ©

, xn пЂ© пѓЋ

d

: 

пѓј xk a couple of п‚Ј 1пѓЅпЂ andпЂ dпЃі n пЂЅ dx1dx2 2 k пЂЅ1 ak пѓѕ

n

Finally, from the result of the multiple integral, we are able to find the formula of the volume of the n-dimensional ellipsoid.

1

Introduction

For the first part of the project, we will see the volumes of prints of 2-dimensional and three-d ellipsoids. This kind of part is performed in order to check for the consistency of the mixture of the volume of n-dimensional ellipsoid. In the process of actually finding the volume of 3-dimensional ellipsoid, we uses the technique of multiple integration plus the Jacobian matrix. After that, we intend to find the formula of the amount of n-dimensional ellipsoid. Even more generally, all of us consider the K function K times, en пЂЅ пѓ• пЂЁ xk a couple of пЂ©. Firstly, if the n ek e пЂЅ1

пЂЁ

пЂ©

exponents are no, then the answer will be the volume of the n-dimensional ellipsoid. Subsequently, if the integrand consists of linear combinations of K features with different exponents, then by linearity of integral, we could calculate the integral simply by separating that into a geradlinig combination of integrals. Finally, we wish to investigate the relation among K function and the likelihood density function of Dirichlet distribution. In order to achieve the objective of finding the formulation, some capabilities should be considered. First, in the lemmas of phase 4, with regard to convenience, two transformations of integral will be considered. In the process of making use of the formula, the thought of reducing aspect will be used. To be able to reduce the aspect, gamma and beta function will be deemed. To evaluate the amount of n-dimensional ellipsoid making use of the formula, several properties of the gamma function should be employed. These real estate are mainly displayed by using the use. After a few process, we all found which the formula may have different expressions for a level integer and and for a strange integer n. Then, using the formula to find the volume of 2-dimensional and 3-dimensional ellipsoid, by checking the outcomes with past calculations, the formula is definitely verified.

2

Chapter one particular Volume of 2-dimensional ellipsoid

To get the volume of a 2-dimensional ellipsoid, it is corresponding to find the area of an ellipse. The formula of an ellipse can be drafted in the form of x2 y a couple of пЂ« пЂЅ1, a two b2 where a and m are the semi-axes of the raccourci.

(1. 1)

There are two methods to locate the area associated with an ellipse: Method I Because the raccourci is symmetric about the x and y responsable, the raccourci is split up into four equal parts by coordinate axes. Hence, the location of an ellipse can be obtained by simply finding the part of a quarter of your ellipse and after that multiplying that by 4. Rewrite (1. 1) in to y пЂЅ п‚±пЂ b one particular пЂ­

x2. Then the uppr part of the ellipse is given by simply a2 x2 y пЂЅ b 1пЂ­ 2 . a

Hence, the spot of the rightmost top corner quarter of the ellipse is just as followings:

x2 пѓІ0 m 1 пЂ­ a a couple of пЂ dx.

a

3

LetпЂ x пЂЅ a sin пЃ±, пЂ thenпЂ dx пЂЅ a cosпЃ± пЂ dпЃ±.

The region of an ellipse

пЂЅпЂ пЂґ пѓІ b one particular пЂ­

0

a

x2 пЂ dx a2

пЂЅ 4ab пѓІ 2 cos 2пЃ± пЂ dпЃ±

0

пЃ°

пЂЅ 2ab пѓІ 2 пЂЁ cos 2пЃ± пЂ« 1пЂ©пЂ dпЃ±

0

пЃ°

пѓ©1 пѓ№2 пЂЅ 2ab пѓЄ desprovisto 2пЃ± пЂ« пЃ± пѓє пѓ«2 пѓ»0 пЂЅ пЃ° ab. вЊ‚ Method 2

пѓ¬ Specify the region E2 пЂЅ пѓ­пЂЁ x, con пЂ© пѓЋ пѓ®

two

пЃ°

:

пѓј x2 y 2 пЂ« 2 п‚Ј 1пѓЅ. 2 a w пѓѕ

Employing double...



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