42 2. The First Variation Since we have only assumed the continuity of fy(x, ˆ ˆ ) and of η (x), this integration by parts is legal. Since η(a) = η(b) = 0 , (2.61) necessary condition (2.58) now reduces to b a ⎛ ⎝ ∂f ∂y − x a ∂f ∂y du⎠ ⎞ ˆ ˆ η (x) dx = 0 . (2.62) We clearly need another lemma to progress further. Here it is: Lemma of du Bois-Reymond: If M(x) ∈ C[a, b] and b a M(x) η (x) dx = 0 (2.63) for all η(x) ∈ C1[a, b] such that η(a) = η(b) = 0 , (2.64) then M(x) = c , (2.65) a constant, for all x ∈ [a, b]. Proof. We may prove this lemma by considering one well-chosen variation η(x). Let μ denote the mean value of M(x) on the closed interval [a, b], μ = 1 (b − a) b a M(x) dx . (2.66) Clearly, b a [M(x) − μ] dx = 0 . (2.67)
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