dKR/dR = u1dlog(f)/dR + u2/R
v2
dJR/dR = vjd log(f )/dR + r Jr.
(10)
The ui, vi, i = 1, 2 are functions of KR, JR, but not explicitly of R itself.
We expand these equations about the critical value KR = KC and about
JR = 0, obtaining
dKR/dR = (KR - KC)ydlog(f)/dR + (KR - KC)z/R
dJR/dR = wJRd log(f)/dR + xJR/R.
(11)
The terms y = du1/dKR|KR=KC, z = du2/dKR|KR=KC, w = v1(KC, 0), x =
v2 (KC, 0) are constants.
Solving the first of these equations gives
KR=KC+(K-KC)Rzf(R)y,
(12)
again remembering that K1 = K, J1 = J, f (1) = 1.
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