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In order to derive the linear relations for solving the singular matrix problem during the recurrence process of the precise integration time-domain method, an electromagnetic closed-surface criterion is presented in orthogonal curvilinear coordinates. Such a criterion leads to the versatile method by which the spatial distributions of the linearly dependent field components are determined. It makes the complicated derivation for linear relation, usually depending on experience, regular, and therefore improves the general-purpose property of the method. With such a criterion, linear relations have been obtained in the computational domain, on the absorbing boundaries and on the edges where boundaries cross in orthogonal curvilinear coordinates. Numerical examples indicate the validity of the criterion.