Curvature and Coriolis

Consider a space constrained by a matrix of functions [M] and containing a physical system represented by a vector array of p-forms |omega>. The Matrix acting as a linear map on |omega> produces a vector array |sigma>. The restricted domain on which the determinant of the matrix is non-zero enables the matrix to be used as a basis Frame for vectors on the space. The existence of an Inverse Frame permits the definition of a linear connection, such that the exterior derivative of any matrix column is a linear combination of the matrix columns.

A second exterior derivative of the Frame will generate the curvature properties of the restricted domain. In effect, the matrix of functions [M] makes a statement about the geometric properties of the restricted space, and in a manner that does not depend upon |omega>.

The map is written symbollically as


The Connection [C] is defined as


Exterior differentiation of the map gives d(map)

[M]{[C]|omega> + d|omega>} = d|sigma>

A second exterior differentiation yields

[M]{[Theta]|omega> + 2[C]|d(omega)> + |dd(sigma)> = |dd(sigma)>

The term [Theta]|omega> represents a Curvature effect.
The term 2[C]|d(omega)> represents a Coriolis effect.
The double exterior derivative terms represent Poincare singularities

IF all functions are C2, the double exterior derviative terms vanish. IT would appear that gravitational effects (due to curvature - [Theta]|omega>) can be balanced out by Coriolis -2[C]|d(omega)>

See the notes in the download below.


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Last update 01/23/2009
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