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Integration Over Spheres and the Divergence Theorem for Balls

where v(s)is the "unit outward normal"to Q at s E AQ andwhere integration on the right is with respect to the "n -1 dimensional surface measure" on AQ. A major obstacle in its formulation is to make sense of the right side of (*), i.e., to suitably define v and integration over AQ. The special case in which Q is a ball (and AQ a sphere) is of considerable interest because, for example, many fundamental properties of harmonic functions (including the mean value property, the weak maximum principle and the Poisson integral formula) follow from it in a relatively painless fashion; see, e.g., Chapter 1 of [1] and Chapter 2 of [4]. Even this case, when included in advanced calculus texts (usually with n = 2 or 3), is either dependent on a sophisticated and lengthy discussion of Stokes' theorem or on a definition of surface integral that involves improper integrals (in dimension n 1), which are usually inadequately treated. Moreover the rotational (or isometr7c) invariance of integration over a sphere in a sense (see the concluding remark) its characteristic property is rarely mentioned in calculus texts. The theory of harmonic functions in two real variables is addressed in many introductory analysis texts, presumably because it is virtually synonymous with analytic function theory (in one complex dimension). Such books are rarely concerned with harmonic functions of three or more variables, perhaps due to the lack of a suitably succinct, but sufficiently profound, version of the divergence theorem. The aim of this article is to develop a utility-grade theory of integration over spheres and use it to formulate and prove the divergence theorem for balls in Rn. Actually, we carry out this programme in detail only in case Q = Bn the closed ball in Rn centered at the origin with radius 1 and where we denote AQ by Sn-l in deference to our topological friends. As we observe in §6, the case of an arbitrary ball can, without difficulty, be reduced to that of Bn. Throughout this paper n denotes a fixed but arbitrary natural number. Unless otherwise indicated, n 2 2. Here is an outline of our development.