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Topic: Differential Forms and Orientation. Case for curves, and
Codimension>1 .

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 Bacle H Posts: 283 Registered: 4/8/12
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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