Sciencia Acta Xaveriana Volume 1 ISSN. 0976-115 No. 1 pp. 101 106 Volume of n-dimensional ellipsoid A. John Wilson Department of Mathematics, Coimbatore Institute of Technology, Coimbatore 641014. India. E - mail: johnwilsonpr@yahoo.com Abstract. In this article the volume of the n-dimensional ellipsoid is derived using the method, step by step process of integration. Recurrence relations are developed to find the volume and surface area of n-dimensional sphere. The relation between the volume and surface area of n-dimensional sphere is given. The asymptotic behavior of the volume and surface area of the unit sphere is also discussed. Key words: n-dimensional ellipsoid, volume, surface area, asymptotic behavior, Euclidean space. Received: 13 December 009) 1 Introduction Let R n be the n-dimensional Euclidean space. The equation to the n-dimensional ellipsoid is given by n x i a i=1 i = 1 where a i denotes the length of the semi axes of the ellipsoid. The ellipsoid axes are aligned with the Cartesian coordinate axes of the n-dimensional Euclidean space. A method to determine the volume of of the ellipsoid is being proposed in this article. This method is based on a step by step process in which the dimension is reduced by one in each step.
10 Volume of n-dimensional ellipsoid Evaluation of volume Theorem.1. The volume of the n-dimensional ellipsoid is given by π n/ V n = n Γn/) a 1,a,...a n ) where a 1,a,...a n, denote the length of semi axes of the n-dimensional ellipsoid. Proof. Let V n be the volume of the n-dimensional ellipsoid n xi a i=1 i V 1 is the volume for x 1 is equal to a 1. V is the area of the ellipse x 1 + x a = 1. 1 a a 1 = 1. = 1, denoting the length of the segment from a 1 to+a 1 which V = πa 1 a = a 1 a β 3, 1 ) V 3 is the volume of the three dimensional ellipsoid x 1 + x a + x 1 a 3 = 1. We can write it as a 3 x 1 ) a1 a 3 x3 ) + x ) a a 3 x3 ) = 1. It is an ellipse with varying semi axes a 1 a 3 x 3 and a a 3 x 3 ranging from to. The required volume is the collection of elliptic plates arranged one above the other. Hence V 3 = π a 1 a a 3 x3 ) dx3 = π a π 1 a 3 cos 3 θdθ, under the substitution x 3 = sinθ 0 3 = a 1 a β, 1 ) β, 1 )
John Wilson 103 V 4 is the volume of the fourth dimensional ellipsoid which can be written as x1 a 1 + x a + x 3 a + x 4 3 a = 1 4 x 1 a1 a 4 ) a 4 x 4 ) + x ) a a a 4 4 x4 ) + x3 ) a3 a a 4 4 x4 ) = 1 It s volume is V 4 = a 1 a 3 β a 4 a 4 a 4, 1 3 = a 1 a a 4 β, 1 ) β, 1 ) a 4 a ) 4 ) β, 1 a 4 x4 ) 3 dx 4 5 β, 1 ) Generalizing this 3 V n = a 1 a a n β, 1 ) β, 1 ) 5 β, 1 ) n β, 1 ) n+1 β, 1 ) π n =a 1 a a n ) n Γn/) on simplification) 3 Special cases The volume of the n-dimensional sphere is π n V n = n Γn/) an, where a is the radius of the sphere. 1) The recurrence formula for the volume of the n-dimensional sphere is given by V n+ = πa n+ V n with V 0 = 1,V 1 = a The concept of volume is meaningful for n 3, for n= it coincides with the common concept of area of a circle and for n=1 with the length of a one dimensional interval
104 Volume of n-dimensional ellipsoid and for n=0, corresponds to a zero-dimensional sphere of volume = 1. On differentiating the volume with respect to the radius we get the surface area. Hence the surface area is S n = dv n da = an 1 π n Γ n / ). ) The recurrence formula for the surface area of the sphere is given by S n+ = πa n S n, n 0 with S 0 = 0 and S 1 =. From 1) and ), the volume and the surface area of n-dimensional sphere are related by V n = a n S n The volume and surface area of the unit sphere is given by π n V n = n Γn/) and S n = π n Γn/) 4 Asymptotic behavior Using the recurrence relations the volume and surface area of the unit sphere for different dimensions are given in Table 1 and are illustrated in Figs. 1 and. It is clear that the volume reaches a maximum for n=5 and the surface area reaches a maximum for n=7 and decays to zero as the dimension increases. 5 Conclusion From the general formula we can find the volume and surface area for any dimension. The volume and surface area for different dimensions are graphically presented. As the dimension is large both volume and surface area asymptotically tends to zero, is an interesting phenomenon.
John Wilson 105 Figure 1: Asymptotic behaviour of volume. Figure : Asymptotic behaviour of surface area.
106 Volume of n-dimensional ellipsoid Table 1: Dimen- Volume Numerical Surface area Numerical sion values values 1.000000 0.00000 π 3.14159 π 06.8319 3 4π / 3 4.188790 4π 1.56637 4 π / 4.93480 π 19.7391 5 8π / 15 5.63789 8π / 3 6.31895 6 π 3/ 6 5.16771 π 3 31.0068 7 16π 3/ 105 4.74765 16π 3/ 15 33.07336 8 π 4/ 4 4.05871 π 4/ 3 3.46969 9 3π 4/ 945 3.98508 3π 4/ 105 9.68658 10 π 5/ 10.550164 π 5/ 1 5.50164 11 64π 5/ 10395 1.884104 64π 5/ 945 0.7514 1 π 6/ 70 1.33563 π 6/ 60 16.0315 13 18π 6/ 135135 0.91068 18π 6/ 10395 11.83817 14 π 7/ 5400 0.59964 π 7/ 360 08.38970 15 56π 7/ 0705 0.381443 56π 7/ 135135 05.7165 16 π 8/ 4030 0.3533 π 8/ 50 03.7659 17 51π 8/ 3445945 0.14098 51π 8/ 0705 0.39668 18 π 9/ 36880 0.0815 π 9/ 0160 01.47863 19 104π 9/ 65479075 0.0466 104π 9/ 3445945 00.88581 0 π 10/ 368800 0.0581 π 10/ 181440 00.51614 References [1] E. Kreyszig, Advanced Engineering Mathematics, Wiley Eastern Limited, 005). [] T. Veerarajan, Engineering Mathematics, Tata McGraw-Hill,006).