About: Calculus of moving surfaces

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The calculus of moving surfaces (CMS) is an extension of the classical tensor calculus to deforming manifolds. Central to the CMS is the Tensorial Time Derivative whose original definition was put forth by Jacques Hadamard. It plays the role analogous to that of the covariant derivative on differential manifolds in that it produces a tensor when applied to a tensor. The Tensorial Time Derivative for a scalar field F defined on is the rate of change in in the instantaneously normal direction: This definition is also illustrated in second geometric figure.

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rdfs:label	Calculus of moving surfaces (en)
rdfs:comment	The calculus of moving surfaces (CMS) is an extension of the classical tensor calculus to deforming manifolds. Central to the CMS is the Tensorial Time Derivative whose original definition was put forth by Jacques Hadamard. It plays the role analogous to that of the covariant derivative on differential manifolds in that it produces a tensor when applied to a tensor. The Tensorial Time Derivative for a scalar field F defined on is the rate of change in in the instantaneously normal direction: This definition is also illustrated in second geometric figure. (en)
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dct:subject	Differential geometry Curvature (mathematics) Calculus Riemannian geometry Tensors
Wikipage page ID	28357925 (xsd:integer)
Wikipage revision ID	1092890571 (xsd:integer)
Link from a Wikipage to another Wikipage	Calculus Product rule Derivative Determinant Velocity Limit of a function Differential geometry Covariance and contravariance of vectors Mathematical analysis Operator (mathematics) Christoffel symbols Function (mathematics) Geometry Levi-Civita symbol Curvature (mathematics) Partial derivative Surface (differential geometry) Curvilinear coordinates Tensor field Jacques Hadamard Covariant derivative Tensor Calculus Riemannian geometry Tensors Chain rule Differentiable manifold Differential geometry Manifold Initial value problem Kronecker delta Metric tensor Rate (mathematics) Geometric Time evolution of integrals Sectional curvature Cartesian coordinate systems Surface normal
sameAs	Calculus of moving surfaces Calculus of moving surfaces Calculus of moving surfaces Calculus of moving surfaces
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has abstract	The calculus of moving surfaces (CMS) is an extension of the classical tensor calculus to deforming manifolds. Central to the CMS is the Tensorial Time Derivative whose original definition was put forth by Jacques Hadamard. It plays the role analogous to that of the covariant derivative on differential manifolds in that it produces a tensor when applied to a tensor. Suppose that is the evolution of the surface indexed by a time-like parameter . The definitions of the surface velocity and the operator are the geometric foundations of the CMS. The velocity C is the rate of deformation of the surface in the instantaneous normal direction. The value of at a point is defined as the limit where is the point on that lies on the straight line perpendicular to at point P. This definition is illustrated in the first geometric figure below. The velocity is a signed quantity: it is positive when points in the direction of the chosen normal, and negative otherwise. The relationship between and is analogous to the relationship between location and velocity in elementary calculus: knowing either quantity allows one to construct the other by differentiation or integration. The Tensorial Time Derivative for a scalar field F defined on is the rate of change in in the instantaneously normal direction: This definition is also illustrated in second geometric figure. The above definitions are geometric. In analytical settings, direct application of these definitions may not be possible. The CMS gives analytical definitions of C and in terms of elementary operations from calculus and differential geometry. (en)
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is Link from a Wikipage to another Wikipage of	Fundamental theorem of calculus Glossary of areas of mathematics Time evolution of integrals The Calculus of Moving Surfaces
is Wikipage redirect of	The Calculus of Moving Surfaces
is foaf:primaryTopic of	wikipedia-en:Calculus_of_moving_surfaces

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