About: First variation of area formula

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In the mathematical field of Riemannian geometry, every submanifold of a Riemannian manifold has a surface area. The first variation of area formula is a fundamental computation for how this quantity is affected by the deformation of the submanifold. The fundamental quantity is to do with the mean curvature. Both sides of the first variation formula can be integrated over S, provided that the variation vector field has compact support. In that case it is immediate from Stokes' theorem that

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rdfs:label	First variation of area formula (en)
rdfs:comment	In the mathematical field of Riemannian geometry, every submanifold of a Riemannian manifold has a surface area. The first variation of area formula is a fundamental computation for how this quantity is affected by the deformation of the submanifold. The fundamental quantity is to do with the mean curvature. Both sides of the first variation formula can be integrated over S, provided that the variation vector field has compact support. In that case it is immediate from Stokes' theorem that (en)
dcterms:subject	Riemannian geometry
Wikipage page ID	16827510 (xsd:integer)
Wikipage revision ID	1122815184 (xsd:integer)
Link from a Wikipage to another Wikipage	Vector field Interior product Lie derivative Compact support Critical point (mathematics) Geodesic Submanifold Stokes' theorem Closed manifold Mean curvature Exterior derivative Partial differential equation Riemannian geometry Riemannian manifold Riemannian metric Riemannian volume form Riemannian geometry Differential form Springer, Cham Surface area Immersion (mathematics) Springer-Verlag Cartan's magic formula Minimal submanifold Real number line
Link from a Wikipage to an external page	https://web.stanford.edu/class/math285/ts-gmt.pdf
sameAs	First variation of area formula First variation of area formula First variation of area formula
dbp:wikiPageUsesTemplate	dbt:Cite_book dbt:Math dbt:Mvar dbt:Short_description
has abstract	In the mathematical field of Riemannian geometry, every submanifold of a Riemannian manifold has a surface area. The first variation of area formula is a fundamental computation for how this quantity is affected by the deformation of the submanifold. The fundamental quantity is to do with the mean curvature. Let (M, g) denote a Riemannian manifold, and consider an oriented smooth manifold S (possibly with boundary) together with a one-parameter family of smooth immersions ft of S into M. For each individual value of the parameter t, the immersion ft induces a Riemannian metric on S, which itself induces a differential form on S known as the Riemannian volume form ωt. The first variation of area refers to the computation in which H(ft) is the mean curvature vector of the immersion ft and Wt denotes the variation vector field Both of these quantities are vector fields along the map ft. The second term in the formula represents the exterior derivative of the interior product of the volume form with the vector field on S, defined as the tangential projection of Wt. Via Cartan's magic formula, this term can also be written as the Lie derivative of the volume form relative to the tangential projection. As such, this term vanishes if each ft is reparametrized by the corresponding one-parameter family of diffeomorphisms of S. Both sides of the first variation formula can be integrated over S, provided that the variation vector field has compact support. In that case it is immediate from Stokes' theorem that In many contexts, S is a closed manifold or the variation vector field is every orthogonal to the submanifold. In either case, the second term automatically vanishes. In such a situation, the mean curvature vector is seen as entirely governing how the surface area of a submanifold is modified by a deformation of the surface. In particular, the vanishing of the mean curvature vector is seen as being equivalent to submanifold being a critical point of the volume functional. This shows how a minimal submanifold can be characterized either by the critical point theory of the volume functional or by an explicit partial differential equation for the immersion. The special case of the first variation formula arising when S is an interval on the real number line is particularly well-known. In this context, the volume functional is known as the length functional and its variational analysis is fundamental to the study of geodesics in Riemannian geometry. (en)
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page length (characters) of wiki page	4729 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:First_variation_of_area_formula
is Link from a Wikipage to another Wikipage of	Mean curvature
is foaf:primaryTopic of	wikipedia-en:First_variation_of_area_formula

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