  We propose a differential operator for computing the residues associated with
a class of meromorphic $n$-forms that frequently appear in the Cachazo-He-Yuan
form of the scattering amplitudes. This differential operator is conjectured to
be uniquely determined by the local duality theorem and the intersection number
of the divisors in the $n$-form. We use the operator to evaluate the tree-level
amplitude of $\phi^3$ theory and the one-loop integrand of Yang-Mills theory
from their CHY forms. The method can reduce the complexity of the calculation.
In addition, the expression for the 1-loop four-point Yang-Mills integrand
obtained in our approach has a clear correspondence with the Q-cut results.
