The Volume And The Surface Area Of A Ellipsoidal CAP

In this thesis we wish to solve two problems.First we will obtain the volume and the surface area of a ellipsoidal cap.The standard equation of n+1-axis ellipsoid in n+1-dimensions Euclidean spaceE_n+1is where a_i＞0,i=1,2,...,n+1.n≥1.Let x_i,(i=1,2,...,n+1)be equal to h(h∈[0,a_i])on x_i-axis,x_i=h is a hyperplane and is a vertical plane of x_i-axis.Since the part of ellipsoidal surface between hyperplane x_i=h and x_i=a_i is said to be the ellipsoidal cap of n+1-axis ellipsoid.Furthermore we will obtain the volume formula of the ellipsoidal cap during hyperplane x₁=a₁ and x₁=h.Consider ellipsoidal cap of n+1-axis ellip-soid x₁²/a₁² + x₂²/a₂² +...+ x_n+1²/a_n+1²â‰¤1 in E_n+1,the part between hyperplane x₁=a₁ and x₁=h,(h∈[0,a₁])is called ellipsoid cap.Let V_n+1(cap)be the volume of the ellipsoid cap.We obtain the volume formula of the ellipsoidal cap where n≥1, I₀=Ï€/2-arcsinh/a_i, I₁=1-h/a₁ V_n+1=Vol(sum from i=1 to n+1(x_i/a_i)²â‰¤1)is the volume of n+1-axis ellipsoid V₀(cap)=1,and V₁(cap)=a₁-h,Γ(.)is the Gamma function. In the third part we get the expression of the surface area of ellipsoid sum from i=1 to n+1 x_i²/a_i²= 1.Let the standard equation of n+1-axis ellipsoid in n+1-dimensions Eu-clidean space be then the area formula of ellipsoidal surface is where...