Note: Numerically, the surface integral of x 1 2 over a unit sphere equals its volume.Įquation (15) for the surface of an n-dimensional sphere of unit radius can be rewritten in terms of the following function σ(x): Surfaces and other surface integrals for n-dimensional spheres with unit radius The value of 2 for n=1, implicit in the above recurrence relations, can be considered as the cardinality of the set of the two end points of a 1D interval. In particular, for n=2 it coincides with the common concept of "circumference" of a circle. This coincides with the popular notion of "surface area" only in the case of n=3 but its extension to any n is quite obvious. The surface of an n-dimensional sphere has the dimension of the (n-1)- st power of length. Showing that the the numbers S n of Eq.(22) are the surfaces of n-dimensional spheres of unit radius whose values can be easily calculated by means of the recurrence When all the p's are zero, Eq.(10) gives simply the surface of the n-dimensional sphere. When only one of the p's equals ν/2 and all the others are zero, Eq.(10) yields the identity, again valid for any ν>-1:įor the important cases of ν=1 and ν=2, this gives In particular, setting ν = 1 and taking into account that Γ(n) = (n-1)!, we obtain the formula Setting all the p's equal to ν/2, one obtains the following formula, valid for any ν>-1: Thus, for example, when p k = -1/4, the corresponding factor in Eq.(3) is the square root of |x k|. Moreover, one should keep in mind that in equations (3) and (5), the individual Cartesian coordinates x k get first squared and only afterwards elevated to p k. Admissible values of the exponents (p's)įor reasons discussed in, the values of p k may assume any real value greater than -1/2, which are also the values compatible with Equation (10). It is therefore pleasing to note that the two procedures lead to the same result.ģ. In the original paper which stimulated this series of Notes, formula (10) was derived in a more direct - and also more laborious - way. Henceįrom which, comparing with (6), we obtain the result ī) Integrals (5) have the nearly self-evident scaling propertyĪrising from the fact that E( r, p) scales with 2p -th power of R and d σ scales with (n-1) -st power or R.Ĭ) For spheres (unlike generic ellipsoids), the volume integration can be carried out by summing the contributions of concentric shells defined by radii r and r+dr, for r ranging from 0 to R. Where d σ is an (n-1)-dimensional surface element.Įvaluation of the integrals (5) is considerably simplified by three facts:Ī) From reference, Equation 17, we know the value of the corresponding volume integral over the n-dimensional sphere, which turns out to be We are interested in the evaluation of the following integrals over S(R): In this notation, an n-dimensional spherical surface S(R) of radius R is defined by the condition The shorthand will be exploited in conventional expressions of the type Which, however, are not to be intended as elements of Rn. We shall often use n-tuples of non-negative real exponents Where the x's are its Cartesian coordinates. Like in, let Rn denote the n-dimensional Euclidean space, r the position vector in Rn and r = | r| its norm We postpone the evaluation of surface integrals over generic ellipsoids because they involve transcendental elliptic integrals. In this Note we focus our attention on a class of surface integrals over n-dimensional real spheres which can be evaluated explicitly. Table of unit spheres surfaces and surface integrals.Next Note in this series: On the Circumference of Ellipses. Previous Note in this series: Volume Integrals over n-Dimensional Ellipsoids. Here we evaluate integrals of products of powers of the Cartesian vector components over surfaces of n-dimensional, real spheres which can be expressed by explicit formulae (this, of course, includes the surface areas themselves). This is the second Note in a series dealing with volume and surface integrals over n-dimensional ellipsoids. Many three-dimensional solids can be generated by revolving a curve about the $x$-axis or $y$-axis.Surface Integrals over n-Dimensional Spheresīy Stanislav Sýkora, Extra Byte, Via R.Sanzio 22C, Castano Primo, Italy 20022
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