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In mathematics, the Beltrami equation, named after Eugenio Beltrami, is the partial differential equation ¯ =. for w a complex distribution of the complex variable z in some open set U, with derivatives that are locally L 2, and where μ is a given complex function in L ∞ (U) of norm less than 1, called the Beltrami coefficient, and where / and / ¯ are Wirtinger derivatives.
Stress functions. In linear elasticity, the equations describing the deformation of an elastic body subject only to surface forces (or body forces that could be expressed as potentials) on the boundary are (using index notation) the equilibrium equation: where is the stress tensor, and the Beltrami-Michell compatibility equations:
The Beltrami identity, named after Eugenio Beltrami, is a special case of the Euler–Lagrange equation in the calculus of variations. The Euler–Lagrange equation serves to extremize action functionals of the form. I a b {\displaystyle I [u]=\int _ {a}^ {b}L [x,u (x),u' (x)]\,dx\,,} where and are constants and . [1]
The Michell solution is a general solution to the elasticity equations in polar coordinates ( ) developed by J. H. Michell. The solution is such that the stress components are in the form of a Fourier series in . Michell [1] showed that the general solution can be expressed in terms of an Airy stress function of the form.
Expressed in terms of components with respect to a rectangular Cartesian coordinate system, the governing equations of linear elasticity are: [1]. Equation of motion: , + = where the (), subscript is a shorthand for () / and indicates /, = is the Cauchy stress tensor, is the body force density, is the mass density, and is the displacement.
Continuum mechanics. In continuum mechanics, a compatible deformation (or strain) tensor field in a body is that unique tensor field that is obtained when the body is subjected to a continuous, single-valued, displacement field. Compatibility is the study of the conditions under which such a displacement field can be guaranteed.
The existence of isothermal coordinates can be proved [6] by applying known existence theorems for the Beltrami equation, which rely on L p estimates for singular integral operators of Calderón and Zygmund. [7] [8] A simpler approach to the Beltrami equation has been given more recently by Adrien Douady. [9] If the Riemannian metric is given ...
Laplace–Beltrami operator. In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named after Pierre-Simon Laplace and Eugenio Beltrami.
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