Orientation sheaf

In algebraic topology, the orientation sheaf on a manifold "X" of dimension "n" is a locally constant sheaf "o""X" on "X" such that the stalk of "o""X" at a point "x" is
(in the integer coefficients or some other coefficients).
Let formula_2 be the sheaf of differential "k"-forms on a manifold "M". If "n" is the dimension of "M", then the sheaf
is called the sheaf of (smooth) densities on "M". The point of this is that, while one can integrate a differential form only if the manifold is oriented, one can always integrate a density, regardless of orientation or orientability; there is the integration map:
If "M" is oriented; i.e., the orientation sheaf of the tangent bundle of "M" is literally trivial, then the above reduces to the usual integration of a differential form.