integration of differential forms
Integration of forms answers: what is the right thing to integrate over a curved space, and over what can you integrate it? The clean answer of this theory is that you integrate a k-form over a k-dimensional oriented (piece of a) manifold. A 1-form integrates over a curve, a 2-form over a surface, an n-form over the whole oriented n-manifold. Forms are precisely the objects designed so that this integral makes sense without any choice of coordinates or metric.
Precisely, to integrate a top-degree n-form omega over an oriented n-manifold M, you first do it in a single chart: there omega = f dx^1 ^ ... ^ dx^n, and you set the integral equal to the ordinary multiple integral of f over the coordinate domain, using an orientation-compatible chart. The reason this is well-defined is the change-of-variables theorem: under an orientation-preserving transition map, the pullback produces exactly the Jacobian determinant, which is precisely the factor the multiple integral needs, so different charts agree. To integrate over all of M, a partition of unity stitches the chart contributions together. More generally, you integrate a k-form over the image of a smooth map from a k-manifold by pulling the form back and integrating downstairs.
This is the payoff of the whole machinery: forms carry their own Jacobians, so integration is automatically coordinate-free, and orientation supplies the sign. It feeds directly into Stokes' theorem and into pairing forms against cycles to compute cohomology. A common misconception: you do NOT need a Riemannian metric to integrate a top-degree form — orientation alone suffices, and the integral of a form is a topological/smooth pairing, not a metric volume. (To integrate a FUNCTION you would need a volume form, hence a metric or a chosen density; integrating a form already includes the 'measure'.)
Integrate omega = x dy over the unit circle, oriented counterclockwise, parameterized by gamma(t) = (cos t, sin t), 0 <= t <= 2 pi. Pull back: gamma^*omega = cos t d(sin t) = cos t (cos t dt) = cos^2 t dt. Then the integral over the circle of omega equals the integral from 0 to 2 pi of cos^2 t dt = pi. (This is the enclosed area, by Green's theorem.) Reverse the orientation, integrate clockwise, and you get -pi — the sign tracks orientation exactly.
Integrating a 1-form over a curve is done by pullback to the parameter interval; orientation reversal flips the sign.
You can only integrate a k-form over a k-dimensional domain, and the domain must be oriented; degree must match dimension. Integrating a form requires no metric, only orientation — a frequent source of confusion when people expect a 'volume' and look for a metric that is not needed.