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Lemma About Derivatives Of Partial Integrals
We conclude this section with the proofs of Theorems 1 and 2. The proof of Theorem 1 uses a lemma about derivatives of partial integrals. LEMMA Suppose P(x, y) is smooth on an open rectangle containing the point (a, b). Then
PROOF The first formula follows at once from the Fundamental Theorem of Calculus. For the second formula, let Δy be a nonzero infinitesimal and let
As y changes to y + Δy, we have
Taking standard parts,
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Last Update: 2010-11-25