On the invariant integration of a vector in some problems in mechanics

Invariant integration of vectors and tensors over manifolds was introduced around fifty years ago by V.N. Folomeshkin, though the concept appears to not have attracted much attention among researchers. Although it is a sophisticated concept, the operation of the invariant integration of vectors is actually required to correctly solve some problems in mechanics. Two such problems are discussed in the present exposition, in the context of a two-dimensional Euclidean space covered by a polar coordinate system. The notion of invariant integration becomes necessary when the space is described without any reference to a Cartesian coordinate system.