Formulation of the Nonorthogonal Div-Curl Problem

Consider a 3-D domain that can be adequately described by the generalized curvilinear coordinate system (u, v, w) and that its mappings are adequately smooth to allow consistent definitions. Then, any vector F can be decomposed into three components with respect to the contravariant a , a , a or the covariant a , a, a,a, linearly independent basis system as 

The quantities f and f are, respectively, the covariant and contravariant components of F which, due to their reciprocity, satisfy the relation a = ht (Sff is the Kronecker s delta). The system s metrical coefficients andg are given by g = g = af a zndg = g = af a, and the determinant of the covariant metrical tensor by [g = %+i (%+2 x a +3),where + 1, + 2, and + 3 denote a consecutive cyclic permutation of coordinates u, v, zu [35]. The last expression reveals that is the Jacobian of the transformation that maps curvilinear 

According to this theoretical framework, one can easily constmct the three fundamental vector operators. So, the gradient of a scalar quantity p is represented by the covariant components as 

Conversely, the divergence of F uses the contravariant components and becomes 

In its general form the div-curl problem reads Let us assume an open bounded domain V, with boundary S. Inside V, there exists a unique vector U, which satisfies 

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