翻訳と辞書 Words near each other ・ Laplace number ・ Laplace operator ・ Laplace operators in differential geometry ・ Laplace plane ・ Laplace pressure ・ Laplace principle (large deviations theory) ・ Laplace transform ・ Laplace transform applied to differential equations ・ Laplace's demon ・ Laplace's equation ・ Laplace's law ・ Laplace's method ・ LaPlace, Louisiana ・ Laplace-P ・ LaplacesDemon ・ Laplace–Beltrami operator ・ Laplace–Carson transform ・ Laplace–Runge–Lenz vector ・ Laplace–Stieltjes transform ・ Laplacian matrix ・ Laplacian of the indicator ・ Laplacian smoothing ・ Laplae District ・ Laplanche ・ LaPlanche Street ・ Lapland ・ Lapland (album) ・ Lapland (electoral district) ・ Lapland (Finland) ・ Lapland (former province of Finland) Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Laplace–Beltrami operator ： ウィキペディア英語版 Laplace–Beltrami operator In differential geometry, the Laplace operator, named after Pierre-Simon Laplace, can be generalized to operate on functions defined on surfaces in Euclidean space and, more generally, on Riemannian and pseudo-Riemannian manifolds. This more general operator goes by the name Laplace–Beltrami operator, after Laplace and Eugenio Beltrami. Like the Laplacian, the Laplace–Beltrami operator is defined as the divergence of the gradient, and is a linear operator taking functions into functions. The operator can be extended to operate on tensors as the divergence of the covariant derivative. Alternatively, the operator can be generalized to operate on differential forms using the divergence and exterior derivative. The resulting operator is called the Laplace–de Rham operator (named after Georges de Rham). The Laplace–Beltrami operator, like the Laplacian, is the divergence of the gradient: :$\backslash nabla^2\; f\; =\; \backslash nabla\; \backslash cdot\; \backslash nabla\; f.$ An explicit formula in local coordinates is possible. Suppose first that ''M'' is an oriented Riemannian manifold. The orientation allows one to specify a definite volume form on ''M'', given in an oriented coordinate system ''x''''i'' by :$\backslash operatorname\_n\; :=\; \backslash sqrt\; \backslash ;dx^1\backslash wedge\; \backslash cdots\; \backslash wedge\; dx^n$ where the ''dxi'' are the 1-forms forming the dual basis to the basis vectors :$\backslash partial\_i\; :=\; \backslash frac$ and $\backslash wedge$ is the wedge product. Here is the absolute value of the determinant of the metric tensor ''g''''ij''. The divergence of a vector field ''X'' on the manifold is then defined as the scalar function with the property :$$ (\nabla \cdot X) \operatorname_n := L_X \operatorname_n where ''LX'' is the Lie derivative along the vector field ''X''. In local coordinates, one obtains :$$ \nabla \cdot X = \frac X^i\right) where the Einstein notation is implied, so that the repeated index ''i'' is summed over. The gradient of a scalar function ƒ is the vector field grad ''f'' that may be defined through the inner product $\backslash langle\backslash cdot,\backslash cdot\backslash rangle$ on the manifold, as :$\backslash langle\; \backslash operatorname\; f(x)\; ,\; v\_x\; \backslash rangle\; =\; df(x)(v\_x)$ for all vectors ''vx'' anchored at point ''x'' in the tangent space ''TxM'' of the manifold at point ''x''. Here, ''d''ƒ is the exterior derivative of the function ƒ; it is a 1-form taking argument ''vx''. In local coordinates, one has :$\backslash left(\backslash operatorname\; f\backslash right)^i\; =$ \partial^i f = g^ \partial_j f where ''gij'' are the components of the inverse of the metric tensor, so that with δ''i''''k'' the Kronecker delta. Combining the definitions of the gradient and divergence, the formula for the Laplace–Beltrami operator applied to a scalar function ƒ is, in local coordinates :$\backslash nabla^2\; f\; =\; \backslash frac\; g^\; \backslash partial\_j\; f\; \backslash right).$ If ''M'' is not oriented, then the above calculation carries through exactly as presented, except that the volume form must instead be replaced by a volume element (a density rather than a form). Neither the gradient nor the divergence actually depends on the choice of orientation, and so the Laplace–Beltrami operator itself does not depend on this additional structure. ==Formal self-adjointness== The exterior derivative ''d'' and −∇ . are formal adjoints, in the sense that for ''ƒ'' a compactly supported function :$\backslash int\_M\; df(X)\; \backslash operatorname\_n\; =\; -\; \backslash int\_M\; f\; \backslash nabla\; \backslash cdot\; X\; \backslash operatorname\_n$     (proof) where the last equality is an application of Stokes' theorem. Dualizing gives for all compactly supported functions ''ƒ'' and ''h''. Conversely, () characterizes the Laplace-Beltrami operator completely, in the sense that it is the only operator with this property. As a consequence, the Laplace–Beltrami operator is negative and formally self-adjoint, meaning that for compactly supported functions ƒ and ''h'', :$\backslash int\_M\; f\backslash ,\backslash nabla^2\; h\; \backslash operatorname\_n\; =\; -\backslash int\_M\; \backslash langle\; d\; f,\; d\; h\; \backslash rangle\; \backslash operatorname\_n\; =\; \backslash int\_M\; h\backslash ,\backslash nabla^2\; f\; \backslash operatorname\_n.$ Because the Laplace–Beltrami operator, as defined in this manner, is negative rather than positive, often it is defined with the opposite sign. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Laplace–Beltrami operator」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.