In vector calculus, and more generally differential geometry, "'Stokes'theorem "'( also called "'the generalized Stokes'theorem "') is a statement about the orientable manifold is equal to the integral of its exterior derivative over the whole of, i . e .,
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On non-orientable manifold,-forms and densities cannot be identified notably, any top-dimensional form must vanish somewhere ( there are no volume forms on non-orientable manifolds ), but there are nowhere-vanishing densities thus while one can integrate densities over compact subsets, one cannot integrate-forms.
23.
On non-orientable manifold,-forms and densities cannot be identified notably, any top-dimensional form must vanish somewhere ( there are no volume forms on non-orientable manifolds ), but there are nowhere-vanishing densities thus while one can integrate densities over compact subsets, one cannot integrate-forms.
24.
A " tensor density " is the special case where " L " is the bundle of " densities on a manifold ", namely the determinant bundle of the cotangent bundle . ( To be strictly accurate, one should also apply the absolute value to the transition functions this makes little difference for an orientable manifold . ) For a more traditional explanation see the tensor density article.
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*An oriented filling of any orientable manifold " X " is another manifold " W " such that the orientation of " X " is given by the boundary orientation of " W ", which is the one where the first basis vector of the tangent space at each point of the boundary is the one pointing directly out of " W ", with respect to a chosen Riemannian metric.
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