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• Laplace–Beltrami_operator abstract "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: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 xi bywhere the dxi are the 1-forms forming the dual basis to the basis vectors and is the wedge product. Here |g| := |det(gij)| is the absolute value of the determinant of the metric tensor gij. The divergence div X of a vector field X on the manifold is then defined as the scalar function with the propertywhere LX is the Lie derivative along the vector field X. In local coordinates, one obtainswhere 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 on the manifold, asfor all vectors vx anchored at point x in the tangent space TxM of the manifold at point x. Here, dƒ is the exterior derivative of the function ƒ; it is a 1-form taking argument vx. In local coordinates, one haswhere gij are the components of the inverse of the metric tensor, so that gijgjk = δik with δik the Kronecker delta.Combining the definitions of the gradient and divergence, the formula for the Laplace–Beltrami operator Δ applied to a scalar function ƒ is, in local coordinatesIf 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.".
• Laplace–Beltrami_operator wikiPageID "1763356".
• Laplace–Beltrami_operator wikiPageRevisionID "597815474".
• Laplace–Beltrami_operator first "E.D.".
• Laplace–Beltrami_operator first "E.V.".
• Laplace–Beltrami_operator id "l/l057450".
• Laplace–Beltrami_operator last "Shikin".
• Laplace–Beltrami_operator last "Solomentsev".
• Laplace–Beltrami_operator title "Laplace–Beltrami equation".
• Laplace–Beltrami_operator subject Category:Differential_operators.
• Laplace–Beltrami_operator subject Category:Riemannian_geometry.
• Laplace–Beltrami_operator comment "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.".
• Laplace–Beltrami_operator label "Laplace–Beltrami operator".
• Laplace–Beltrami_operator label "Operador de Laplace-Beltrami".
• Laplace–Beltrami_operator label "Operador de Laplace-Beltrami".
• Laplace–Beltrami_operator label "Operatore di Laplace-Beltrami".
• Laplace–Beltrami_operator label "Opérateur de Laplace-Beltrami".
• Laplace–Beltrami_operator label "Оператор Лапласа — Бельтрами".
• Laplace–Beltrami_operator label "拉普拉斯－贝尔特拉米算子".
• Laplace–Beltrami_operator sameAs Laplace%E2%80%93Beltrami_operator.
• Laplace–Beltrami_operator sameAs Operador_de_Laplace-Beltrami.
• Laplace–Beltrami_operator sameAs Opérateur_de_Laplace-Beltrami.
• Laplace–Beltrami_operator sameAs Operatore_di_Laplace-Beltrami.
• Laplace–Beltrami_operator sameAs 라플라스-벨트라미_연산자.
• Laplace–Beltrami_operator sameAs Operador_de_Laplace-Beltrami.
• Laplace–Beltrami_operator sameAs Q1071846.
• Laplace–Beltrami_operator sameAs Q1071846.
• Laplace–Beltrami_operator wasDerivedFrom Laplace–Beltrami_operator?oldid=597815474.