Functions of bounded higher variation

We say that a function u: ℝ^m → ℝⁿ, with m ≥ n, has bounded n-variation if Det(u_xα1,...,u_xαn) is a measure for every 1 ≤ α₁ <...< α_n ≤ m. Here Det(v₁,...,v_n) denotes the distributional determinant of the matrix whose columns are the given vectors, arranged in the given order. In this paper we establish a number of properties of BnV functions and related functions. We establish general (and rather weak) versions of the chain rule and the coarea formula; we show that stronger forms of the chain rule can fail, and we also demonstrate that BnV functions cannot, in general, be strongly approximated by smooth functions; and we prove that if u ∈ BnV(ℝ^m,ℝⁿ) and |u| = 1 a.e., then the Jacobian of u is an m - n-dimensional rectifiable current.