Calculus of moving surfaces

Short description: Extension of the classical tensor calculus

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 ∇α on differential manifolds in that it produces a tensor when applied to a tensor.

Jacques Salomon Hadamard, French Mathematician, 1865–1963 CE

Suppose that Σt is the evolution of the surface Σ indexed by a time-like parameter t. The definitions of the surface velocity C and the operator ∇˙ are the geometric foundations of the CMS. The velocity C is the rate of deformation of the surface Σ in the instantaneous normal direction. The value of C at a point P is defined as the limit

C=limh->0Distance(P,P*)h

where P* is the point on Σt+h that lies on the straight line perpendicular to Σt at point P. This definition is illustrated in the first geometric figure below. The velocity C is a signed quantity: it is positive when PP*‾ points in the direction of the chosen normal, and negative otherwise. The relationship between Σt and C 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 δ/δt-derivative of an invariant field F

The Tensorial Time Derivative ∇˙ for a scalar field F defined on Σt is the rate of change in F in the instantaneously normal direction:

δFδt=limh->0F(P*)−F(P)h

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.

## Contents

* 1 Analytical definitions
* 2 Properties of the ∇˙-derivative
* 3 Differentiation table for the ∇˙-derivative
* 4 Time differentiation of integrals
* 5 References

## Analytical definitions

For analytical definitions of C and ∇˙, consider the evolution of S given by

Zi=Zi(t,S)

where Zi are general curvilinear space coordinates and Sα are the surface coordinates. By convention, tensor indices of function arguments are dropped. Thus the above equations contains S rather than Sα. The velocity object V=ViZi is defined as the partial derivative

Vi=∂Zi(t,S)∂t

The velocity C can be computed most directly by the formula

C=ViNi

where Ni are the covariant components of the normal vector N→.

Also, defining the shift tensor representation of the Surface's Tangent Space Ziα=Sα⋅Zi and the Tangent Velocity as Vα=ZiαVi , then the definition of the ∇˙ derivative for an invariant F reads

∇˙F=∂F(t,S)∂t−Vα∇αF

where ∇α is the covariant derivative on S.

For tensors, an appropriate generalization is needed. The proper definition for a representative tensor Tjβiα reads

∇˙Tjβiα=∂Tjβiα∂t−Vη∇ηTjβiα+VmΓmkiTjβkα−VmΓmjkTkβiα+Γ˙ηαTjβiη−Γ˙βηTjηiα

where Γmjk are Christoffel symbols and Γ˙βα=∇βVα−CBβα is the surface's appropriate temporal symbols (Bβα 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

∇˙(SαiTjβ)=Tjβ∇˙Sαi+Sαi∇˙Tjβ

and obeys a chain rule for surface restrictions of spatial tensors:

∇˙Fkj(Z,t)=∂Fkj∂t+CNi∇iFkj

Chain rule shows that the ∇˙-derivatives of spatial "metrics" vanishes

∇˙δji=0,∇˙Zij=0,∇˙Zij=0,∇˙εijk=0,∇˙εijk=0

where Zij and Zij are covariant and contravariant metric tensors, δji is the Kronecker delta symbol, and εijk and εijk 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 Zij.

## 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 Sαβ and the contravariant metric tensor Sαβ, the following identities result

∇˙Sαβ=0∇˙Sαβ=0

where Bαβ and Bαβ are the doubly covariant and doubly contravariant curvature tensors. These curvature tensors, as well as for the mixed curvature tensor Bβα, satisfy

∇˙Bαβ=∇α∇βC+CBαγBβγ∇˙Bβα=∇β∇αC+CBγαBβγ∇˙Bαβ=∇α∇βC+CBγαBγβ

The shift tensor Zαi and the normalNi satisfy

∇˙Zαi=Ni∇αC∇˙Ni=−Zαi∇αC

Finally, the surface Levi-Civita symbols εαβ and εαβ satisfy

∇˙εαβ=0∇˙εαβ=0

## Time differentiation of integrals

The CMS provides rules for time differentiation of volume and surface integrals.

## References

1. ↑ 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.
2. ↑ J. Hadamard, Leçons Sur La Propagation Des Ondes Et Les Équations de l'Hydrodynamique. Paris: Hermann, 1903.

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