Using direct methods in the calculus of variations, \textit{G. Anzellotti, R. Serapioni}, and \textit{I. Tamanini} [Indiana Univ. Math. J. 39, 617-669 (1990; Zbl 0718.49030)] considered the problem of minimizing functionals \(\mathcal F (M)\), defined on a surface \(M\) and depending on the curvatures of \(M\). The key idea was to consider the graph \(G\) of the Gauss map of a surface \(M\), and to consider functionals \(\mathcal F\) such that Area \((G)\leq c \mathcal F(M)\). In [Boll. Unione Mat. Ital., VII. Ser., B 10, 991-1017 (1996; Zbl 0886.49031)] the present author introduced the notion of a generalized Gauss graph for any codimension in terms of integer multiplicity rectifiable currents, in particular, the Gauss graphs of smooth oriented submanifolds. In this paper, the author proves the divergence theorem, derives the first variation formula for generalized Gauss graphs, and studies some relations between mean curvature measure and first variation of the nonoriented varifold associated to a generalized Gauss graph.