Integration on a manifold requires a coherent choice of sign between coordinate systems. Orientation supplies that sign, and partitions of unity then reduce the global integral to ordinary multivariable integration.
Orientation
Definition. An orientation of an \(n\)-manifold is a smooth choice of one of the two orientation classes of ordered bases of \(T_pM\). Equivalently, it is an atlas whose transition maps have positive Jacobian determinant.
Proposition. A smooth \(n\)-manifold is orientable if and only if it admits a nowhere-vanishing \(n\)-form.
Proof. A nonzero top form declares a basis positive when its value on that basis is positive. Conversely, choose positive coordinate top forms and patch them using a partition of unity. Because transition determinants are positive, the local forms have compatible sign, and their weighted sum never vanishes.
The M?bius strip is nonorientable: transporting a transverse frame once around its central circle reverses orientation. Every connected orientable manifold has exactly two orientations.
Integration of top forms
Definition. If \(\omega\) is a compactly supported \(n\)-form contained in a positively oriented chart \((U,x)\), with \(\omega=f\,\mathrm{d}x^1\wedge\cdots\wedge\mathrm{d}x^n\), set \[ \int_M\omega=\int_{x(U)}f(x^{-1}(y))\,\mathrm{d}y. \] For general compact support, choose an oriented partition of unity and sum the chart integrals.
Proposition (change of variables). If \(F:M\to N\) is an orientation-preserving diffeomorphism and \(\omega\) is a compactly supported top form, then \[ \int_MF^*\omega=\int_N\omega. \]
Proof. A partition of unity reduces the assertion to one coordinate neighborhood. There it is the Euclidean change-of-variables theorem; the positive Jacobian removes the absolute-value ambiguity.
Boundary orientation
If \(M\) is oriented and has boundary, the induced orientation on \(\partial M\) follows the outward-normal-first convention: \((v_1,\ldots,v_{n-1})\) is positive when \((\nu_{\mathrm{out}},v_1,\ldots,v_{n-1})\) is positive in \(M\). For an interval \([a,b]\), this makes \(b\) positive and \(a\) negative, recovering \(\int_{\partial[a,b]}f=f(b)-f(a)\).
Densities
A density transforms by the absolute value of the Jacobian and can be integrated without orientation. A Riemannian metric produces the density \[ \mathrm{d}V_g=\sqrt{\det(g_{ij})}\, |\mathrm{d}x^1\cdots\mathrm{d}x^n|. \] On an oriented Riemannian manifold it corresponds to the volume form \(\operatorname{vol}_g=\sqrt{\det g}\, \mathrm{d}x^1\wedge\cdots\wedge\mathrm{d}x^n\).