Concept# Curl (mathematics)

Summary

In vector calculus, the curl, also known as rotor, is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional Euclidean space. The curl at a point in the field is represented by a vector whose length and direction denote the magnitude and axis of the maximum circulation. The curl of a field is formally defined as the circulation density at each point of the field.
A vector field whose curl is zero is called irrotational. The curl is a form of differentiation for vector fields. The corresponding form of the fundamental theorem of calculus is Stokes' theorem, which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve.
The notation curl F is more common in North America. In many European countries, particularly in classic scientific literature, the alternative notation rot F is traditionally used, which comes from the "rate of rotation" that i

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In this thesis, we study two distinct problems.
The first problem consists of studying the linear system of partial differential equations which consists of taking a k-form, and applying the exterior derivative 'd' to it and add the wedge product with a 1-form 'a'. The study of this differential operator is linked to the study of the multiplication by a two form, that is the system of linear equations where we take a k-form and apply the exterior wedge product by 'da', the exterior derivative of 'a'. We establish links between the partial differential equation and the linear system.
The second problem is a generalization of the symmetric gradient and the curl equation. The equation of a symmetric gradient consists of taking a vector field, apply the gradient and then add the transpose of the gradient, whereas in the curl equation we subtract the transpose of the gradient. Both can be seen as an equation of the form A * grad u + (grad u)t * A, where A is a symmetric matrix for the case of the symmetric gradient and skew symmetric for the curl equation. We generalize to the case where A verifies no symmetry assumption and more significantly add a Dirichlet condition on the boundary.

In this thesis, we study some linear and nonlinear problems involving differential forms. We begin by studying the problem of pullbacks which asks the following question: for two given differential forms, if one is the pullback of the other via a diffeomorphism satisfying some given condition. For volume forms, this problem was studied by Dacorogna-Moser giving a necessary and sufficient condition for the existence of the diffeomorphism with precise regularity. Our goal is to extend this result for general k-forms. We have obtained some necessary and sufficient conditions for two-forms and for some special classes of k-forms with sharp regularity. Then we turn our attention to the problem of differential inclusions involving differential forms. Although for zero-forms, the problem has been extensively studied, essentially nothing was known for higher forms including the curl operator. In this direction, we have obtained some necessary and some sufficient conditions for general k-forms unifying the study of the different cases. Moreover, we show that these necessary and sufficient conditions coincide for k = 1, solving the case of curl operator fairly completely. Besides these problems, we have studied some domain invariance property of the weighted-homogenous and non-homogenous Hardy constants as well. We have showed that the Hardy constant corresponding to these classes of inequalities enjoy, to some extent, the same domain invariance property as that of the Hardy constant corresponding to the standard Hardy's inequality.