Abstract We present a detailed description of the recent idea for a direct decomposition of Feynman integrals onto a basis of master integrals by projections. as well as a direct derivation of the differential equations satisfied by the master integrals. employing multivariate intersection numbers. We discuss a recursive algorithm for the computation of multivariate intersection numbers. https://parisnaturalfoodes.shop/product-category/white-chestnut/