The surface of a flag in the wind is an example of a deforming manifold.
The calculus of moving surfaces (CMS) [1] is an extension of the classical tensor calculus to deforming manifolds. Central to the CMS is the Tensorial Time Derivative
whose original definition [2] 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.
Jacques Salomon Hadamard, French Mathematician, 1865–1963 CE
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
![C=\lim _{h\to 0}{\frac {{\text{Distance}}(P,P^{*})}{h}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98f905ed5e0f9fb4e4d5979bf6d41a4d42bdb719)
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.
Geometric construction of the surface velocity C
Geometric construction of the
![\delta /\delta t](https://wikimedia.org/api/rest_v1/media/math/render/svg/b574fe3b20b3430800864fa4a11cc72c229dc23a)
-derivative of an invariant field F
The Tensorial Time Derivative
for a scalar field F defined on
is the rate of change in
in the instantaneously normal direction:
![{\frac {\delta F}{\delta t}}=\lim _{h\to 0}{\frac {F(P^{*})-F(P)}{h}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cb3b0f03c7bd664a1ee2b9ce2150bd1817021d2)
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.
Analytical definitions[]
For analytical definitions of
and
, consider the evolution of
given by
![{\displaystyle Z^{i}=Z^{i}\left(t,S\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/946537679703d03a4dda47463d95dd435ca0c25b)
where
are general curvilinear space coordinates and
are the surface coordinates. By convention, tensor indices of function arguments are dropped. Thus the above equations contains
rather than
. The velocity object
is defined as the partial derivative
![{\displaystyle V^{i}={\frac {\partial Z^{i}\left(t,S\right)}{\partial t}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea9e267664ba7aa3e34ef4c5afcab97d7c002262)
The velocity
can be computed most directly by the formula
![{\displaystyle C=V^{i}N_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2746aa03930a808bcbdaadf4ab19f0e4c5ac845c)
where
are the covariant components of the normal vector
.
Also, defining the shift tensor representation of the Surface's Tangent Space
and the Tangent Velocity as
, then the definition of the
derivative for an invariant F reads
![{\displaystyle {\dot {\nabla }}F={\frac {\partial F\left(t,S\right)}{\partial t}}-V^{\alpha }\nabla _{\alpha }F}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7c8b2358816b1f95707f98a7f4accfb138fc1d4)
where
is the covariant derivative on S.
For tensors, an appropriate generalization is needed. The proper definition for a representative tensor
reads
![{\displaystyle {\dot {\nabla }}T_{j\beta }^{i\alpha }={\frac {\partial T_{j\beta }^{i\alpha }}{\partial t}}-V^{\eta }\nabla _{\eta }T_{j\beta }^{i\alpha }+V^{m}\Gamma _{mk}^{i}T_{j\beta }^{k\alpha }-V^{m}\Gamma _{mj}^{k}T_{k\beta }^{i\alpha }+{\dot {\Gamma }}_{\eta }^{\alpha }T_{j\beta }^{i\eta }-{\dot {\Gamma }}_{\beta }^{\eta }T_{j\eta }^{i\alpha }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e67d51fb5f7d3f46804bed634f5ef9605e17a55d)
where
are Christoffel symbols and
is the surface's appropriate temporal symbols (
is a matrix representation of the surface's curvature shape operator)
Properties of the
-derivative[]
The
-derivative commutes with contraction, satisfies the product rule for any collection of indices
![{\displaystyle {\dot {\nabla }}(S_{\alpha }^{i}T_{j}^{\beta })=T_{j}^{\beta }{\dot {\nabla }}S_{\alpha }^{i}+S_{\alpha }^{i}{\dot {\nabla }}T_{j}^{\beta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2afe3fef9275419b63894f7dfaafa41e6780a9e4)
and obeys a chain rule for surface restrictions of spatial tensors:
![{\displaystyle {\dot {\nabla }}F_{k}^{j}(Z,t)={\frac {\partial F_{k}^{j}}{\partial t}}+CN^{i}\nabla _{i}F_{k}^{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b193ccb6f30a91d66b8eb6169254455683d25de)
Chain rule shows that the
-derivatives of spatial "metrics" vanishes
![{\displaystyle {\dot {\nabla }}\delta _{j}^{i}=0,{\dot {\nabla }}Z_{ij}=0,{\dot {\nabla }}Z^{ij}=0,{\dot {\nabla }}\varepsilon _{ijk}=0,{\dot {\nabla }}\varepsilon ^{ijk}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f17c2b149abad0183ebb8b47b99367c1dc17029b)
where
and
are covariant and contravariant metric tensors,
is the Kronecker delta symbol, and
and
are the Levi-Civita symbols. The 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
.
Differentiation table for the
-derivative[]
The
derivative of the key surface objects leads to highly concise and attractive formulas. When applied to the covariant surface metric tensor
and the contravariant metric tensor
, the following identities result
![{\displaystyle {\begin{aligned}{\dot {\nabla }}S_{\alpha \beta }&=0\\[8pt]{\dot {\nabla }}S^{\alpha \beta }&=0\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a5f6e7f6ec7eae3edd762cb8941e78157ec50ef)
where
and
are the doubly covariant and doubly contravariant curvature tensors. These curvature tensors, as well as for the mixed curvature tensor
, satisfy
![{\displaystyle {\begin{aligned}{\dot {\nabla }}B_{\alpha \beta }&=\nabla _{\alpha }\nabla _{\beta }C+CB_{\alpha \gamma }B_{\beta }^{\gamma }\\[8pt]{\dot {\nabla }}B_{\beta }^{\alpha }&=\nabla _{\beta }\nabla ^{\alpha }C+CB_{\gamma }^{\alpha }B_{\beta }^{\gamma }\\[8pt]{\dot {\nabla }}B^{\alpha \beta }&=\nabla ^{\alpha }\nabla ^{\beta }C+CB^{\gamma \alpha }B_{\gamma }^{\beta }\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f2e2b5072dab70f45f0a4d67650ce163479475c)
The shift tensor
and the normal
satisfy
![{\displaystyle {\begin{aligned}{\dot {\nabla }}Z_{\alpha }^{i}&=N^{i}\nabla _{\alpha }C\\[8pt]{\dot {\nabla }}N^{i}&=-Z_{\alpha }^{i}\nabla ^{\alpha }C\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04583f21c0d1c60912dadb938116e2c2c65a548d)
Finally, the surface Levi-Civita symbols
and
satisfy
![{\displaystyle {\begin{aligned}{\dot {\nabla }}\varepsilon _{\alpha \beta }&=0\\[8pt]{\dot {\nabla }}\varepsilon ^{\alpha \beta }&=0\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9dace09f2d05c3613013ee0373e46cfbbfe808fa)
Time differentiation of integrals[]
The CMS provides rules for time differentiation of volume and surface integrals.
References[]
- ^ Grinfeld, P. (2010). "Hamiltonian Dynamic Equations for Fluid Films". Studies in Applied Mathematics. doi:10.1111/j.1467-9590.2010.00485.x. ISSN 0022-2526.
- ^ J. Hadamard, Leçons Sur La Propagation Des Ondes Et Les Équations de l'Hydrodynamique. Paris: Hermann, 1903.