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By Brian D O'Neill, Mark Styling

The 1st identify within the Elite devices sequence to house an American bombardment workforce, this name specializes in the 303rd BG, dubbed the 'Hells Angels.' one of many first actual B-17 devices assigned to the newly created 8th Air strength in England in September 1942, the 303rd used to be within the forefront of the sunlight bombing crusade via to VE-Day. offered a unique Unit quotation in January 1944, the 303rd additionally had of its aircrewmen provided with the Medal of Honor, Americas final army ornament. Brian O Neill brings the group's vibrant strive against historical past to existence with a mixture of first-hand money owed, uncooked records and concise project narrative.

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The map → dχ (x, u(x))z is a vector-field along the path γ : → χ (x, u(x)). We find dχ (x, u(x))z = χ (x, u(x))i,j z j ∂xi |τ (x) . We abbreviate χ = χ (x, u(x)) and ψ = ψ (x, u(x)). By the formula for the covariant derivative along γ we have d (dχ z) d dχi ,j j i z + Γjk χj,l z l γ˙ k ∂xi |χ = d i = ξ(χ , ψ )i,j z j + Γjk χj,l z l ξ k (χ , ψ ) ∂xi |χ j i l j i α j i l k = ξ,x l (χ , ψ )χ ,j z + ξ,uα (χ , ψ )ψ ,j z + Γjk χ ,l z ξ (χ , ψ ) ∂xi |χ . Collecting terms this can be written as d (dχ z) = (Dx ξ)(χ , ψ )dχ z + (∂uα ξ)(χ , ψ ) ⊗ dψ α z d = (Dx ξ)(χ , ψ ) + (∂uα ξ)(χ , ψ ) ⊗ dψ α ◦ dχ−1 dχ z.

Similarly the partial divergence divx ξ(x, u) is defined. Total derivatives Suppose u : M → Rk is a smooth Rk -valued function. For f : M × Rk → R we consider the new real-valued function f (Id ×u)(x) := f (x, u(x)). For the total gradient of f (Id ×u) : M → R we have the formula ∇(f (x, u(x))) = ∇x f (x, u(x)) + ∂uα f (x, u(x))∇uα , i written in local coordinates as f ;i ∂xi = (f ;x + f,uα uα;i )∂xi . A similar constructions leads to the total differential of f (Id ×u): d(f (x, u(x))) = dx f (x, u(x)) + ∂uα f (x, u(x))duα i k written in local coordinates as f,i dxi = (f,xi +f,uα uα ,i )dx .

10) for a large LM > 0. First, this restricts the result to all those solutions which attain values in [−M, M ]. But since M can be taken arbitrarily large the full statement is recovered. For u ∈ W01,2 (B1 (0)) let uj := B1 (0) uψj dx for j ≥ 1. Let g u := ∞ − 2 2 j=1 e (u − u0 )j ψj + P (u − u0 ) + u0 , where P : L (B1 (0)) → Lrad is the orthogonal projection. Consider the functional L[u] = B1 (0) 12 |∇u|2 −F (x, u) dx on W01,2 (B1 (0)) with F (x, s) = s 0 f (x, t) dt. 5 Classical uniqueness results d L[g u] d ∞ =0 =− ∞ µj (u − u0 )j uj + j=1 By (a) we know µj u0,j = d L[g u] d =0 d L[g u] d =0 =− µj − ∞ j=1 ∞ ∞ (f (x, u)−f (x, u0 ))(u−u0 )j ψj dx.

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