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...Symbolic tensor calculus on manifolds eric gourgoulhon laboratoire univers et theories luth cnrs observatoire de paris universite diderot sciences lettres meudon france http sagemanifolds obspm fr jncf journees nationales calcul formel centre international rencontres mathematiques luminy marseille january jan outline introduction smooth scalar elds vector conclusion and perspectives pseudo riemannian musical isomorphisms levi civita connection curvature hodge duality what is by it usually meant arithmetics of product contraction anti symmetrization lie derivative along a eld pullback pushforward associated to manifold map exterior dierential forms...