Generalized Jacobians are related to the degree theory of mappings \(u:\Omega\to\mathbb{R}^n\), where \(\Omega\subset \mathbb{R}^n\) is an open subset. We can state here only the following (rather technical) main result. Let \(g:\mathbb{R}^m\to S^{n-1}\) be a mapping of finite fractional order Sobolev norm \(g\in W^{1-1/n,n}(\mathbb{R}^m,S^{n-1})<\infty\) and let \(u\in W^{1,n}_{\text{loc}}(\mathbb{R}^{m+1}_+,\mathbb{R}^n)\) be an extension of \(g\) such that \(u|_{\mathbb{R}^m}=g\). Then the formula \[ \langle\text{Sing}(g),\tau\rangle=\frac{1}{\omega_n}\int_{\mathbb{R}^{m+1}_+} d\widetilde{\tau}\wedge u^*(dy^1\wedge\dots \wedge dy^n) \] (where \(\tau\) is a variable \((m-n)\)-form on \(\mathbb{R}^m\) with compact support and \(\widetilde{\tau}\) is a smooth \((m-n)\)-form on \(\mathbb{R}^{m+1}\) such that \(\widetilde{\tau}|_{\mathbb{R}^m}=\tau)\) determines an \((m-n)\)-current \(\text{Sing}(g)\). Theorem: \(\partial J=\text{Sing}(g)\) for appropriate integer multiplicity \((m-n+1)\)-current \(J\) and the mass of \(J\) can be estimated by the norm of \(g\).