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We describe higher-order chain rules for multivariate functions and tensorfields. We estimate Sobolev-Slobodeckij norms, Musielak-Orlicz norms, and thetotal variation seminorms of the higher derivatives of tensor fields after achange of variables and determine sufficient regularity conditions for thecoordinate change. We also introduce a novel higher-order chain rule forcomposition chains of multivariate functions that is described via nested setpartitions and generalized Bell polynomials; it is a natural extension of theFa\`a di Bruno formula. Our discussion uses the coordinate-free language oftensor calculus and includes Fr\'echet-differentiable mappings between Banachspaces.

Higher-order chain rules for tensor fields, generalized Bell polynomials, and estimates in Orlicz-Sobolev-Slobodeckij and bounded variation spaces