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Beltrami Vector Field
In vector calculus, a Beltrami vector field, named after Eugenio Beltrami, is a vector field in three dimensions that is parallel to its own curl (mathematics), curl. That is, F is a Beltrami vector field provided that \mathbf\times (\nabla\times\mathbf)=0. Thus \mathbf and \nabla\times\mathbf are parallel vectors in other words, \nabla\times\mathbf = \lambda \mathbf. If \mathbf is solenoidal - that is, if \nabla \cdot \mathbf = 0 such as for an incompressible fluid or a magnetic field, the identity \nabla \times (\nabla \times \mathbf) \equiv -\nabla^2 \mathbf + \nabla (\nabla \cdot \mathbf) becomes \nabla \times (\nabla \times \mathbf) \equiv -\nabla^2 \mathbf and this leads to -\nabla^2 \mathbf = \nabla \times(\lambda \mathbf) and if we further assume that \lambda is a constant, we arrive at the simple form \nabla^2 \mathbf = -\lambda^2 \mathbf. Beltrami vector fields with nonzero curl correspond to Euclidean contact forms in three dimensions. The vector field \mathbf = -\fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Calculus
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used extensively in physics and engineering, especially in the description of electromagnetic fields, gravitational fields, and fluid flow. Vector calculus was developed from quaternion analysis by J. Willard Gibbs and Oliver Heaviside near the end of the 19th century, and most of the notation and terminology was established by Gibbs and Edwin Bidwell Wilson in their 1901 book, ''Vector Analysis''. In the conventional form using cross products, vector calculus does not generalize to higher dimensions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curl (mathematics)
In vector calculus, the curl 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. is a notation common today to the United States and Americas. In many European countries, particularly in classic scientific literature, the alternative notation is traditionally used, which is spelled as "rotor", and comes from the "rate of rotation", which it rep ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contact Form
In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution may be given (at least locally) as the kernel of a differential one-form, and the non-integrability condition translates into a maximal non-degeneracy condition on the form. These conditions are opposite to two equivalent conditions for ' complete integrability' of a hyperplane distribution, i.e. that it be tangent to a codimension one foliation on the manifold, whose equivalence is the content of the Frobenius theorem. Contact geometry is in many ways an odd-dimensional counterpart of symplectic geometry, a structure on certain even-dimensional manifolds. Both contact and symplectic geometry are motivated by the mathematical formalism of classical mechanics, where one can consider either the even-dimensional phase space of a mechanical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beltrami Flow
In fluid dynamics, Beltrami flows are flows in which the vorticity vector \mathbf and the velocity vector \mathbf are parallel to each other. In other words, Beltrami flow is a flow where Lamb vector is zero. It is named after the Italian mathematician Eugenio Beltrami due to his derivation of the Beltrami vector field, while initial developments in fluid dynamics were done by the Russian scientist Ippolit S. Gromeka in 1881. Description Since the vorticity vector \boldsymbol and the velocity vector \mathbf are parallel to each other, we can write :\boldsymbol\times\mathbf=0, \quad \boldsymbol = \alpha(\mathbf,t) \mathbf, where \alpha(\mathbf,t) is some scalar function. One immediate consequence of Beltrami flow is that it can never be a planar or axisymmetric flow because in those flows, vorticity is always perpendicular to the velocity field. The other important consequence will be realized by looking at the incompressible vorticity equation :\frac + (\mathbf\cdot\nabla)\bo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Lamellar Vector Field
In vector calculus, a complex lamellar vector field is a vector field which is orthogonal to a family of surfaces. In the broader context of differential geometry, complex lamellar vector fields are more often called hypersurface-orthogonal vector fields. They can be characterized in a number of different ways, many of which involve the curl. A lamellar vector field is a special case given by vector fields with zero curl. The adjective "lamellar" derives from the noun "lamella", which means a thin layer. The ''lamellae'' to which "lamellar vector field" refers are the surfaces of constant potential, or in the complex case, the surfaces orthogonal to the vector field. This language is particularly popular with authors in rational mechanics. Complex lamellar vector fields In vector calculus, a complex lamellar vector field is a vector field in three dimensions which is orthogonal to its own curl. That is, :\mathbf\cdot (\nabla\times \mathbf) = 0. The term lamellar vector fiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conservative Vector Field
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of any path between two points does not change the value of the line integral. Path independence of the line integral is equivalent to the vector field under the line integral being conservative. A conservative vector field is also irrotational; in three dimensions, this means that it has vanishing curl. An irrotational vector field is necessarily conservative provided that the domain is simply connected. Conservative vector fields appear naturally in mechanics: They are vector fields representing forces of physical systems in which energy is conserved. For a conservative system, the work done in moving along a path in a configuration space depends on only the endpoints of the path, so it is possible to define potential energy that is independent of the actual path taken. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
