THE METHOD OF STATIONARY PHASE 11 The eigenvectors of (^-(w)) are called the principal directions of curvature of M in the direction n. The numbers Ki = fx~l are called the principal radii of curvature. If we set px(y) = \x — y\ so that \j x = \[yx] then the Hessians satisfy Hp = X -H*. In our case M is two dimensional so that 1 \y — x\ \y — x\ |det #p|1/2 |det H^12 |(1 - | , - x\h)(\ - | , - x\^f2' For each x and y let d denote the distance from x to y. If x — y is normal to M let # denote the number focal points between y and x. Substituting into our original integral gives 4 = T 2 e^*™2^ ^ ^ + 0(*"2). * y\E(yj,)-x \{\ - lixd){\ ^ li2d)\V2 So far we have been considering the rather artificial situation where each point on a surface radiates uniformly in all directions. We will now show that similar considerations apply to physically interesting solutions of the wave equation, with the real surface replaced by an imaginary surface and use made of Stokes' theorem. We begin by recalling Green's formula: Let u and v be two functions on R3 then d(u * dv — v * du) = ud * dv — vd * du so, by Stokes' theorem J if ukv - vAudV = f((u*dv-v* du), D dD if D is any bounded region with smooth boundary. If u satisfies Aw 4- k u = 0 and v satisfies Lv + k2v = 8P then the left hand side becomes u(P), if P G D and 0 if P £ D. Thus we have the formula of Helmholtz ±//[£--"(S)H: (P) if P E A ^H L r - " " V r ) \ ~ {0 if P £ A if w is a solution of (A2 + /c2)w = 0 and r denotes the distance from P. In many applications we are interested in the situation where D, instead of being bounded, represents the exterior to some surface S. Let us first apply the formula to the bounded region DR, consisting of the intersection of D with a ball of radius R centered at P.

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