Can you help for understanding this definitions in a good way.What is the my problem is that I can not image this definition and proposition in my mind and also ı dont understand the reason that ı write gren writing.Also what is actually $i_{x}w$.These are basi thinks but can someone give me light these definition and propostiton,I will so happy thanks
The orientation form is defined as an every where non-zero n form.
This means that its value on any n linearlly independent vectors is always non-zero.
The boundary of the manifold has a unit inward point normal vector. At each point of the boundary the unit normal can be extended to n linearly independent vectors by adding any n-1 linearly independent vectors that are tangent to the boundary. The orientation form on the manifold is never zero on these n frames and thus defines by contraction with the unit normal an orientation form on the boundary