  Manifold calculus of functors, due to M. Weiss, studies contravariant
functors from the poset of open subsets of a smooth manifold to topological
spaces. We introduce "multivariable" manifold calculus of functors which is a
generalization of this theory to functors whose domain is a product of
categories of open sets. We construct multivariable Taylor approximations to
such functors, classify multivariable homogeneous functors, apply this
classification to compute the derivatives of a functor, and show what this
gives for the space of link maps. We also relate Taylor approximations in
single variable calculus to our multivariable ones.
