User:Hh73wiki/The Calculus of Moving Surfaces Update
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The calculus of moving surfaces (CMS) [1] is an extension of the classical tensor calculus to deforming manifolds. Central to the CMS is the -derivative whose original definition [2] was put forth by Pavel Grinfeld. It plays the role analogous to that of the covariant derivative
on differential manifolds. In particular, it has the property that it produces a tensor when applied to a tensor.
![Thumb image](http://upload.wikimedia.org/wikipedia/commons/thumb/8/83/Dannebrog.jpg/640px-Dannebrog.jpg)
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.
![Thumb image](http://upload.wikimedia.org/wikipedia/commons/thumb/4/45/C3x3t.png/640px-C3x3t.png)
![Thumb image](http://upload.wikimedia.org/wikipedia/commons/thumb/0/02/DFdt3x3t.png/640px-DFdt3x3t.png)
The -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.