Dr. Katz DEq Homework Solutions 56

Dr. Katz DEq Homework Solutions 56 - 56 N.M. Katz: P roof A...

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56 N.M. Katz: Proof According to (3.5.1.2), we have (3.5.5.1) Lie (P)(a (v)) = dr(v) whence for any integer my > 1, we have (3.5.5.2) Lie (p)mv (a (v)) = Lie (p)mv-1 (df(v)) = d (pm~-I (f(v))). Let us expand the second member of (3.5.5.0), remembering that the a(v), and hence the Lie(P)m~(a(v)), are closed forms. d(P m'-l(f(1)) ~ (- 1) i+1 m~ em'-~(f(i)) 1-[ Lie(p)mv(a(v))) i*l v*l,i (3.5.5.3) =Lie(p)m'(a(1))~(-1)i+lmipm'-l(f(i)) ~I Lie(p)mv(a(v)) i#l + e='-'(f(1)) E (- 1) '+1 rn, Lie(P)"'(a(i)) 1-I Lie(p)m"(a(v))" i.1 The first term in (3.5.5.4) is the part of the first member of (3.5.5.0) corre- sponding to i. 1. Thus it remains to see that ml P"'-x(f(1)) I-I Lie(p)m'(a(v)) (3.5.5.4) v*l = P"'- '(f(1)) Z (- 1) i+~ m, Lie (P)"'(a(i)) 1-I Lie (P)m~(a
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