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Topic:
Differential Forms and Orientation. Case for curves, and
Codimension>1 .
Replies:
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Bacle H
Posts:
283
Registered:
4/8/12
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Differential Forms and Orientation. Case for curves, and
Codimension>1 .
Posted:
Nov 8, 2012 9:35 PM
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Hi, All:
One way of defining an orientation form for an codimension-1 ,
orientable n-manifold N embedded in R^{n+1} , in which
the gradient ( of the parametrized image ) is non-zero, is to
consider the nowhere-zero normal vector n(x), and to define the
form w(v)_x : = | n(x) v1 , v2 ,...,v_n-1| (##)
Where {vi}_i=1,..,n-1 is an orthogonal basis for T_x N , written
as column vectors, and n(x) is the vector normal to N at x .
Then the vectors in (##) are pairwise orthogonal, and so are
Linearly-independent.
*QUESTION* : How do we define a form for a curve of codimension-1,
and, in general, for orientable manifolds of codimension larger-
than 1 ?
Thanks.
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