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Dr. Katz DEq Homework Solutions 50

# Dr. Katz DEq Homework Solutions 50 - 50 N.M Katz(3.5.1.3...

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50 N.M. Katz: (3.5.1.3) (3.5.1.4) (3.5.1.5) (3.5.1.6) Proof simply calculate if ie~, Lie(Q)(a(/)) =0, I(P-Q)(z(i))=g(i), I(P-Q)(a(i))=f(i), I(P p - Qp) (a (i)) = h (i). (3.5.1.1) is by definition (3.5.0.1) of ~ As for (3.5.0.2), we Lie(P)(a(i))=Lie(e) ( dx, I = Lie(e)(dxi) P(x,) dx~ \ xi ] (3.5.1.7) dP(xi) P(xi) dxi X i X i X i =d \-~-i !=tiT(i)" (3.5.1.8) if ir we multiply the above calculation (3.5.1.7) by x~. (3.5.1.3) holds because Q (xi)= 0 by definition (3.5.0) of Q. As for (3.5.1.4), we calculate (3.5.1.9) if ie~, liP- Q)(z(i))= I(P- Q) ( dxi ] - (P- Q) (xi) - P(x,) =g(i), \ xi ! (3.5.1.1o) if ir l(P-Q)(z(i))=I(P-Q)(dxO=(P-Q)(xi)=P(x,)=g(i) and (3.5.1.5) follows similarly:
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