en>RWyn: /* Overview */

2014-01-16T17:52:02Z

Overview

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[[File:Dannebrog.jpg|thumb|200px|The surface of a flag in the wind is an example of a deforming manifold.]]

The '''calculus of moving surfaces''' ('''CMS''') <ref>Grinfeld, P. (2010). "Hamiltonian Dynamic Equations for Fluid Films". Studies in Applied Mathematics. {{doi|10.1111/j.1467-9590.2010.00485.x}}. ISSN 00222526.</ref> is an extension of the classical [[Tensor|tensor calculus]] to deforming [[manifold]]s. Central to the CMS is the <math>\delta /\delta t</math>-[[derivative]] whose original definition <ref>J. Hadamard, Lecons Sur La Propagation Des Ondes et Les Equations De
l’Hydrodynamique. Paris: Hermann, 1903.</ref> was put forth by [[Jacques Hadamard]]. It plays the role analogous to that of the [[covariant derivative]] <math>\nabla _{\alpha }</math> on [[Differentiable manifold|differential manifolds]]. In particular, it has the property that it produces a [[tensor]] when applied to a tensor.

[[File:Hadamard2.jpg|thumb|right|thumb|130px|Jacques Salomon Hadamard, French Mathematician, 1865–1963 CE]]

Suppose that <math>S_t</math> is the evolution of the [[surface]] <math>S</math> indexed by a time-like parameter <math>t</math>. The definitions of the surface [[velocity]] <math>C</math> and the [[operator (mathematics)|operator]] <math>\delta /\delta t</math> are the [[geometric]] foundations of the CMS. The velocity C is the [[rate (mathematics)|rate]] of deformation of the surface <math>S</math> in the instantaneous [[Surface normal|normal]] direction. The value of <math>C</math> at a point <math>P</math> is defined as the [[limit of a function|limit]]

: <math>C=\lim_{h\to 0} \frac{\text{Distance}(P,P^*)}{h}</math>

where <math>P^{*}</math> is the point on <math>S_{t+h}</math> that lies on the straight line perpendicular to <math>S_{t}</math> at point P. This definition is illustrated in the first geometric figure below. The velocity <math>C</math> is a signed quantity: it is positive when <math>\overline{PP^{*}}</math> points in the direction of the chosen normal, and negative otherwise. The relationship between <math>S_{t}</math> and <math>C</math> is analogous to the relationship between location and velocity in elementary calculus: knowing either quantity allows one to construct the other by [[Derivative|differentiation]] or [[Initial value problem|integration]].

[[File:C3x3t.png|thumb|350px|Geometric construction of the surface velocity C]]
[[File:DFdt3x3t.png|thumb|350px|Geometric construction of the <math>\delta/\delta t</math>-derivative of an invariant field F]]

The <math>\delta /\delta t</math>-derivative for a scalar field F defined on <math>S_{t}</math> is the [[derivative|rate of change]] in <math>F</math> in the instantaneously normal direction:

: <math>\frac{\delta F}{\delta t}=\lim_{h\to 0} \frac{F(P^*)-F(P)}{h}</math>

This definition is also illustrated in second geometric figure.

The above definitions are ''[[geometry|geometric]]''. In analytical settings, direct application of these definitions may not be possible. The CMS gives ''analytical'' definitions of C and <math>\delta /\delta t</math> in terms of elementary operations from [[calculus]] and [[differential geometry]].

==Analytical definitions==

For [[mathematical analysis|analytical]] definitions of <math>C</math> and <math>\delta /\delta t</math>, consider the evolution of <math>S</math> given by

: <math>Z^i = Z^i \left( t ,S \right) \, </math>

where <math>Z^{i}</math> are general [[curvilinear coordinates|curvilinear space coordinates]] and <math>S^{\alpha }</math> are the surface coordinates. By convention, tensor indices of function arguments are dropped. Thus the above equations contains <math>S</math> rather than <math>S^\alpha</math>.The velocity object
<math>v^{i}</math> is defined as the [[partial derivative]]

: <math>v^i =\frac{\partial Z^i \left( t ,S \right)}{\partial t }</math>

The velocity <math>C</math> can be computed most directly by the formula

: <math>C=v^i N_i \, </math>

where <math>N_i</math> are the covariant components of the normal vector <math>\vec{N}</math>.

The definition of the <math>\delta /\delta t</math>-derivative for an [[Tensor field|invariant]] ''F'' reads

: <math>\frac{\delta F}{\delta t}=\frac{\partial F\left( t ,S \right)}{\partial t }-v^{i}Z^{\alpha }_{i}\nabla _{\alpha }F</math>

where <math>Z^\alpha_i</math> is the shift tensor and
<math>\nabla_\alpha </math> is the covariant derivative on S.

For ''tensors'', an appropriate generalization is needed. The proper definition for a representative tensor <math>T^{i\alpha }_{j\beta }</math> reads

: <math>\frac{\delta T^{i\alpha }_{j\beta }}{\delta t}=\frac{\partial T^{i\alpha }_{j\beta }}{\partial t}-v^{\eta }\nabla _{\eta }T^{i\alpha }_{j\beta }+v^{m}\Gamma ^{i}_{mk}T^{k\alpha }_{j\beta }-v^{m}\Gamma ^{k}_{mj}T^{i\alpha }_{k\beta }+\nabla _{\eta }v^{\alpha }T^{i\eta }_{j\beta }-\nabla _{\beta }v^{\eta }T^{i\alpha }_{j\eta }</math>

where <math>\Gamma^k_{mj}</math> are [[Christoffel symbols]].

==Properties of the ''δ''/''δt''-derivative==

The <math>\delta /\delta t</math>-derivative commutes with contraction, satisfies the [[product rule]] for any collection of indices

: <math>\frac{\delta }{\delta t}\left( S^i_\alpha T^\beta_j
\right)=\frac{\delta S^i_\alpha}{\delta t}T^\beta_j + S^i_\alpha \frac{\delta T^\beta_j }{\delta t}</math>

and obeys a [[chain rule]] for surface [[Function_(mathematics)#Restrictions_and_extensions|restrictions]] of spatial tensors:

: <math>\frac{\delta F^j_k}{\delta t} =\frac{\partial F^j_k}{\partial t}+CN^i \nabla _i F^j_k</math>

Chain rule shows that the <math>\delta /\delta t</math>-derivative of spatial "metrics"
vanishes

: <math>\frac{\delta \delta^i_j}{\delta t},\frac{\delta Z_{ij}}{\delta t},\frac{\delta Z^{ij}}{\delta t},\frac{\delta \varepsilon _{ijk}}{\delta t},\frac{\delta \varepsilon^{ijk}}{\delta t}=0</math>

where
<math>Z_{ij}</math> and <math>Z^{ij}</math> are covariant and contravariant [[metric tensor]]s, <math>\delta ^{i}_{j}</math> is the [[Kronecker delta]] symbol, and <math>\varepsilon _{ijk}</math> and <math>\varepsilon ^{ijk}</math> are the [[Levi-Civita symbol]]s. The [[Levi-Civita symbol|main article]] on Levi-Civita symbols describes them for [[Cartesian coordinate systems]]. The preceding rule is valid in general coordinates, where the definition of the Levi-Civita symbols must include the square root of the [[determinant]] of the covariant metric tensor
<math>Z_{ij}</math>.

==Differentiation table for the δ/δt-derivative==

The <math>\delta /\delta t</math>-derivative of the key surface objects leads to highly concise and attractive formulas. When applied to the [[covariance and contravariance of vectors|covariant]] surface [[metric tensor]] <math>S_{\alpha \beta }</math> and the [[Covariance and contravariance of vectors|contravariant]] metric tensor
<math>S^{\alpha \beta }</math>, the following identities result

: <math>\begin{align}
\frac{\delta S_{\alpha \beta }}{\delta t} & = -2CB_{\alpha \beta } \\[8pt]
\frac{\delta S^{\alpha \beta }}{\delta t} & = 2CB^{\alpha \beta }
\end{align}</math>

where <math>B_{\alpha \beta }</math> and <math>B^{\alpha \beta }</math> are the doubly covariant and doubly contravariant [[Sectional curvature|curvature tensors]]. These curvature tensors, as well as for the mixed curvature tensor <math>B^\alpha_\beta</math>, satisfy

: <math>\begin{align}
\frac{\delta B_{\alpha \beta }}{\delta t}& = \nabla _\alpha \nabla_\beta C - CB_{\alpha \gamma }B^\gamma_\beta \\[8pt]
\frac{\delta B^\alpha_\beta}{\delta t}& = \nabla^\alpha \nabla_\beta C + CB^\alpha_\gamma B^\gamma_\beta \\[8pt]
\frac{\delta B^{\alpha \beta }}{\delta t}& = \nabla ^\alpha \nabla^\beta C + 3CB^\alpha_\gamma B^{\gamma \beta}
\end{align}</math>

The shift tensor <math>Z^i_\alpha</math> and the normal
<math>N^i</math> satisfy

: <math>\begin{align}
\frac{\delta Z^i_\alpha}{\delta t} & = \nabla _\alpha \left( CN^i \right) \\[8pt]
\frac{\delta N^i}{\delta t} & = -Z^i_\alpha \nabla^\alpha C
\end{align}</math>

Finally, the surface [[Levi-Civita symbol]]s <math>\varepsilon _{\alpha \beta }</math> and <math>\varepsilon ^{\alpha \beta }</math> satisfy

: <math>\begin{align}
\frac{\delta \varepsilon _{\alpha \beta }}{\delta t} & = -\varepsilon _{\alpha \beta }CB^{\gamma }_{\gamma } \\[8pt]
\frac{\delta \varepsilon ^{\alpha \beta }}{\delta t} & = \varepsilon ^{\alpha \beta }CB^\gamma_\gamma
\end{align}</math>

== Time differentiation of integrals ==
The CMS provides rules for [[time evolution of integrals|time differentiation of volume and surface integrals]].

==References==
{{Reflist}}
[[Category:Tensors]]
[[Category:Differential geometry]]
[[Category:Riemannian geometry]]
[[Category:Curvature_(mathematics)]]
[[Category:Calculus]]

Wins Above Replacement - Revision history

en>RWyn: /* Overview */