For a bounded domain Omega subset of R-N with connected Lipschitz boundary, we prove the existence of some c > 0, such thatc parallel to P parallel to(L2(Omega,RNxN)) <= parallel to symP parallel to(L2(Omega,RNxN)) + parallel to CurlP parallel to(L2(Omega,RNx(N-1)N/2))holds for all square-integrable tensor fields P : Omega -> R-NxN, having square-integrable generalized "rotation" Curl P : Omega -> RNx(N-1)N/2 and vanishing tangential trace on partial derivative Omega, where both operations are to be understood row-wise. Here, in each row, the operator curl is the vector analytical reincarnation of the exterior derivative d in RN. For compatible tensor fields P, that is, P = del v, the latter estimate reduces to a non-standard variant of Korn's first inequality in R-N, namelyfor all vector fields v is an element of H-1 (Omega, R-N), for which del v(n), n = 1, ... , N, are normal at partial derivative Omega. Copyright (C) 2011 John Wiley & Sons, Ltd.c parallel to del v parallel to(L2(Omega,RNxN)) <= parallel to sym del v parallel to(L2(Omega, RNxN))