  Nonlocal and fractional-order models capture effects that classical partial
differential equations cannot describe; for this reason, they are suitable for
a broad class of engineering and scientific applications that feature
multiscale or anomalous behavior. This has driven a desire for a vector
calculus that includes nonlocal and fractional gradient, divergence and
Laplacian type operators, as well as tools such as Green's identities, to model
subsurface transport, turbulence, and conservation laws. In the literature,
several independent definitions and theories of nonlocal and fractional vector
calculus have been put forward. Some have been studied rigorously and in depth,
while others have been introduced ad-hoc for specific applications. The goal of
this work is to provide foundations for a unified vector calculus by (1)
consolidating fractional vector calculus as a special case of nonlocal vector
calculus, (2) relating unweighted and weighted Laplacian operators by
introducing an equivalence kernel, and (3) proving a form of Green's identity
to unify the corresponding variational frameworks for the resulting nonlocal
volume-constrained problems. The proposed framework goes beyond the analysis of
nonlocal equations by supporting new model discovery, establishing theory and
interpretation for a broad class of operators, and providing useful analogues
of standard tools from the classical vector calculus.
