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Examples of Boundary Orientations
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For H^n with orientation form dx1 wedge ... dxn, using the outward pointing vector field -d/dxn we get that the boundary orientation is (-1)^n dx1 wedge.... wedge dx_{n-1}. For [a,b] with orientation dx, using -d/dx at a and d/dx at b we get orientation at a is -1 and at b is 1. For image of smooth immersion from [a,b] to M, we use differential to induce an orientation.
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