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Integration and orientation

A differential k -form can be integrated over an oriented manifold of dimension k. A differential 1 -form can be thought of as measuring an infinitesimal oriented length, or 1-dimensional oriented density. A differential 2 -form can be thought of as measuring an infinitesimal oriented area, or 2-dimensional oriented density. And so on.

Integration of differential forms is well-defined only on oriented manifolds. An example of a 1-dimensional manifold is an interval, and intervals can be given an orientation: they are positively oriented if a < b, and negatively oriented otherwise. If a < b then the integral of the differential 1 -form f (x) dx over the interval (with its natural positive orientation) is

which is the negative of the integral of the same differential form over the same interval, when equipped with the opposite orientation. That is:

This gives a geometrical context to the conventions for one-dimensional integrals, that the sign changes when the orientation of the interval is reversed. A standard explanation of this in one-variable integration theory is that, when the limits of integration are in the opposite order (b < a), the increment dx is negative in the direction of integration.

More generally, an m -form is an oriented density that can be integrated over an m -dimensional oriented manifold. (For example, a 1 -form can be integrated over an oriented curve, a 2 -form can be integrated over an oriented surface, etc.) If M is an oriented m -dimensional manifold, and M ′ is the same manifold with opposite orientation and ω is an m -form, then one has:

These conventions correspond to interpreting the integrand as a differential form, integrated over a chain. In measure theory, by contrast, one interprets the integrand as a function f with respect to a measure μ and integrates over a subset A, without any notion of orientation; one writes ∫ A f d μ = ∫ f d μ {\textstyle \int _{A}f\,d\mu =\int _{}f\,d\mu } to indicate integration over a subset A. This is a minor distinction in one dimension, but becomes subtler on higher-dimensional manifolds; see below for details.

Making the notion of an oriented density precise, and thus of a differential form, involves the exterior algebra. The differentials of a set of coordinates, dx,..., dx can be used as a basis for all 1 -forms. Each of these represents a covector at each point on the manifold that may be thought of as measuring a small displacement in the corresponding coordinate direction. A general 1 -form is a linear combination of these differentials at every point on the manifold:

where the f = f (x,..., x) are functions of all the coordinates. A differential 1 -form is integrated along an oriented curve as a line integral.

The expressions dx ∧ dx, where i < j can be used as a basis at every point on the manifold for all 2 -forms. This may be thought of as an infinitesimal oriented square parallel to the x – x -plane. A general 2 -form is a linear combination of these at every point on the manifold: ∑ 1 ≤ i < j ≤ n f i, j d x i ∧ d x j {\textstyle \sum _{1\leq i<j\leq n}f_{i,j}\,dx^{i}\wedge dx^{j}}, and it is integrated just like a surface integral.

A fundamental operation defined on differential forms is the exterior product (the symbol is the wedge ∧). This is similar to the cross product from vector calculus, in that it is an alternating product. For instance,

because the square whose first side is dx and second side is dx is to be regarded as having the opposite orientation as the square whose first side is dx and whose second side is dx. This is why we only need to sum over expressions dx ∧ dx, with i < j; for example: a (dx ∧ dx) + b (dx ∧ dx) = (a − b) dx ∧ dx. The exterior product allows higher-degree differential forms to be built out of lower-degree ones, in much the same way that the cross product in vector calculus allows one to compute the area vector of a parallelogram from vectors pointing up the two sides. Alternating also implies that dx ∧ dx = 0, in the same way that the cross product of parallel vectors, whose magnitude is the area of the parallelogram spanned by those vectors, is zero. In higher dimensions, dx ∧ ⋅⋅⋅ ∧ dx = 0 if any two of the indices i,..., i are equal, in the same way that the "volume" enclosed by a parallelotope whose edge vectors are linearly dependent is zero.