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Understanding asymptotics of gradient components in relation to the symmetrized gradient is important for the analysis of buckling of slender structures. For circular cylindrical shells we obtain the exact scaling exponent of the Korn constant as a function of shell's thickness. Equally sharp results are obtained for individual components of the gradient in cylindrical coordinates. We also derive an analogue of the Kirchhoff ansatz, whose most prominent feature is its singular dependence on the slenderness parameter, in marked contrast with the classical case of plates and rods. [ABSTRACT FROM AUTHOR]