THE METHOD OF STATIONARY PHASE 9 y. Furthermore, the index of d \px, i.e., the number of negative eigenvalues of this quadratic form, is given by ind(rfV ) = 2 c o r a n k dE (y w) 0/i v ^ ' This result is a special case of the Morse index theorem. Loosely speaking, the formula says that the index of d\j/x is the number of focal points between y and x, counted with multiplicity. Let us introduce local coordinates as above, and let F: Ux R w - / - Rm be the map E expressed in terms of these coordinates thus F(u,s) = y{u) + s n{u) = y(u) + sMnM{u) + + smnm(u). Thus 3—*~ ZJ sr^—, i = 1, . . . , m, nr, r = / + 1, . . . , m. Now the point (y, v) G N(M) whose coordinates are (u,s) will be a regular point for E if and only if the m vectors dF dUj' dsr are linearly independent. The m vectors (dy/du(), nr are always linearly independ- ent by construction. Taking the scalar product with the (dF/du^, (dF/dsr) we get the matrix 0 2 S r ( ^ n ' ) ! ! which is non-singular if and only if the upper left hand block is non-singular. With no loss of generality we may assume that we have chosen nt+] = n to point in the direction of v so that v sn with s 0. Thus the corank of dEx v \ is the same as the corank of the matrix Now A.( *y.\ = (*i *L\ ( dl y \ 3M,. V ' duj) \ 3«, , duJ) + V"' 3u. 3w. ) - dF 3«,.:
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