36 2. The First Variation Let us now look at the total variation ΔJ = J( ) − J(0) . (2.31) That is, ΔJ = b a f(x, ˆ + η, ˆ + η ) dx − b a f(x, ˆ ˆ ) dx (2.32) = b a [f(x, ˆ + η, ˆ + η ) − f(x, ˆ ˆ )] dx . If f has enough continuous partial derivatives — and we shall assume that it does — we may expand the total variation in a power series in . Using the usual Taylor expansion, we obtain ΔJ = δJ + 1 2 δ2J + O( 3 ) . (2.33) Here, δJ = dJ( ) d = 0 (2.34) = b a [fy(x, ˆ ˆ ) η + fy (x, ˆ ˆ ) η ] dx is the first variation. Likewise, δ2J = d2J( ) d2 = 0 2 (2.35) = 2 b a [fyy(x, ˆ ˆ ) η2 + 2 fyy (x, ˆ ˆ ) η η + fy y (x, ˆ ˆ ) η 2 ] dx is the second variation. For sufficiently small, we expect that a nonvanishing first variation will dominate the right-hand side of total variation (2.33). Likewise, we expect that a nonvanishing second variation will dominate higher-order terms. If J[ˆ] is a relative (or local) minimum, we must have ΔJ ≥ 0 (2.36)
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