Abstract. Variational sequence of order $r$ introduced by Krupka is defined on the $r$-jet prolongation of a fibered manifold $\pi :Y\rightarrow X$, as a quotient sequence of the de Rham sequence of sheaves. In this paper we describe explicitly the elements of the 1st-order variational sequence over an $n$-dimensional manifold $X$.
AMSclassification. 58E99, 58A12, 58A15, 58A20
Keywords. Fibered manifold, contact form, strongly contact form, trace decomposition problem, $q$-equation, sheaf, variational sequence