On tensor functions whose gradients have some skew-symmetries

Montanaro, Adriano

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 2 (1991) no. 3, pp. 259-268

Résumé

Let \( \mathcal{V}_{n} \) be a real inner product space of any dimension; and let \( Q^{\alpha_{1} \dots \alpha_{v}} = \hat{Q}^{\alpha_{1} \dots \alpha_{v}} (X_{\beta{1} \dots \beta_{\tau}}) \) be a \( C^{2} \)-map relating any two tensor spaces on \( \mathcal{V}_{n} \). We study the consequences imposed on the form of this function by the condition that its gradient should be skew-symmetric with respect to some pairs \( ( \alpha_{\mu},\beta_{\eta} ) \) of indexes. Any such a condition is written as a system of linear partial differential equations, with constant coefficients, which is symmetric with respect to certain couples of independent variables. The solutions of these systems appear useful to characterize the possible indéterminations in the admissible systems of constitutive equations for various continuous media.