We are interested in the numerical reconstruction of a vector field with prescribed divergence and curl in a general 3 or 2 dimensional domain, not necessarily contractible. To this aim, we introduce some basic concepts of finite element exterior calculus and rely heavily on recent results of P. Leopardi and A. Stern. The goal of the paper is to take advantage of the links between usual vector calculus and exterior calculus and show the interest of the exterior calculus framework, without too much prior knowledge of the subject. We start by describing the method used for contractible domains and its implementation using the FEniCS library (see fenicsproject.org). We then address the problems encountered with non contractible domains and general boundary conditions and explain how to adapt the method to handle these cases. Finally we give some numerical results obtained with this method, in dimension 2 and 3.