This communication revisits the algebra-based results for derivative estimation presented by Fliess and coauthors in 2005. It is proposed, here, to consider multidimensional functions, namely scalar or vector fields of several variables. Such fields are locally represented by a vector Taylor series expansion, and a computation technique is presented so to put successive partial derivatives (for instance, the gradient, the Hessian matrix...) as functions of iterated integrals of the measured quantities.