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1 Y Cao, G Sheng, J X Zhang, S Choudhury, Y L Li, C Randall, and L Q Chen Piezoelectric response of singlecrystal PbZr 1x Ti x O 3 near morphotropicX X k k j X X j n k n −= j j = ≤ = ∑,α α, , ,α 1 0 for 1 Since X r and rr X k k k n,α α = ∑ 1 have equal coordinates, they must be the same vector ♥ This theorem gives us a convenient and useful representation for an arbitrary vector Also, the proof illustrates an important technique for showing that two vectors are equalH j = x j x j 1 be 'moment' at jth point Then between x Citeseerx Ist Psu Edu Viewdoc Download Doi 10 1 1 476 4863 Rep Rep1 Type Pdf ƒ~ƒƒJƒƒX ƒMƒƒƒ‰ƒhƒX