A fundamental task in discrete differential geometry is the choice of representation of differential quantities. Different representations will lead to different optimization problems when using DDG in practice, and can considerably influence the scope of feasible applications. We will describe a novel choice of representation, based on functional operators, which although standard in classic differential geometry has only been recently applied to DDG. We will show how operators which take scalar functions to scalar functions can be used to concisely represent smooth maps between shapes and smooth vector fields. We further demonstrate that by using a common operator representation, the intimate connection between maps and vector fields can be leveraged to easily compute vector fields which fulfill intricate global constraints, such as Killing and symmetric vector fields. Finally, we show how the spectral decomposition of the Laplace-deRham operator can be leveraged for combining local and global vector field constraints in a unified framework.