Pullbacks and Integration on Parametrized Domains

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Integration over a parametrized oriented domain is defined by pulling the form back to ordinary Euclidean coordinates. An orientation-preserving reparametrization leaves the integral unchanged because its Jacobian determinant appears twice in complementary roles: in the pulled-back top form and in the change of variables. An orientation-reversing reparametrization changes the sign.

Required background. Tensor Fields and Natural Operations — use smooth maps, pullbacks, tangent vectors, and covariant tensor transformation laws.

Helpful background. Calculus and Linear Algebra for Relativity — use multiple integrals, Jacobian determinants, and the Euclidean change-of-variables theorem.

Pulling a form to its parameter domain

Let $\sigma:D\to M$ be a smooth parametrization of an oriented $k$ -dimensional region, with coordinates $u^1,\ldots,u^k$ on $D\subset\mathbb R^k$ . For a $k$ -form $\omega$ on $M$ , define

\int_{\sigma(D)}\omega \equiv \int_D\sigma^*\omega,

provided the parametrization represents the region with the intended multiplicity and the integral converges. In coordinates $x^i$ on $M$ ,

(\sigma^*\omega)_{a_1\cdots a_k} = \omega_{i_1\cdots i_k}(\sigma(u)) \frac{\partial\sigma^{i_1}}{\partial u^{a_1}} \cdots \frac{\partial\sigma^{i_k}}{\partial u^{a_k}}.

This formula says to evaluate $\omega$ on the tangent vectors carried by the parametrization. It does not require a metric or an induced area element. The construction and its relationship to ordinary multivariable integration are developed in Guillemin and Haine 2019, §4.5, pp. 142–146, PDF.

Reparametrization invariance and orientation

Suppose $h:D'\to D$ is a diffeomorphism. The reparametrized map is $\sigma'=\sigma\circ h$ , and functoriality gives

(\sigma')^*\omega=h^*(\sigma^*\omega).

If $h$ preserves orientation, the ordinary change-of-variables theorem yields

\int_{D'}h^*(\sigma^*\omega)=\int_D\sigma^*\omega.

If $h$ reverses orientation, the oriented integral acquires a minus sign. This is not an arbitrary convention: the determinant in the pullback retains its sign, unlike the absolute Jacobian used for integrating scalar functions against positive measure.

For general parametrizations, one must also track multiplicity. A map that covers a surface twice gives twice the integral. A map whose differential drops rank can fail to represent a regular oriented patch even though its pullback remains algebraically defined. Reparametrization invariance applies to orientation-compatible diffeomorphisms of parameter domains, not to arbitrary many-to-one maps.

Polar coordinates on the disk

Let $\sigma(r,\theta)=(r\cos\theta,r\sin\theta)$ for $0\lt r\lt R$ and $0\lt\theta\lt2\pi$ . Then

\sigma^*(dx\wedge dy)=r\,dr\wedge d\theta.

Although polar coordinates degenerate at $r=0$ and duplicate a seam, these sets have measure zero; one may integrate over truncated sectors and take a limit. The result is

\int_{D_R}dx\wedge dy =\int_0^{2\pi}\int_0^R r\,dr\,d\theta =\pi R^2.

Now reverse the angular parameter with $h(r,\theta)=(r,2\pi-\theta)$ . Since $h^*(dr\wedge d\theta)=-dr\wedge d\theta$ , the same geometric disk is traversed with the opposite orientation and the integral changes to $-\pi R^2$ . Replacing the signed form by a density would remove this sign, a distinction developed on the orientation page.

Curves, surfaces, and limits of notation

For a curve $\gamma:[a,b]\to M$ and a one-form $A=A_i dx^i$ ,

\int_\gamma A =\int_a^b A_i(\gamma(t))\dot\gamma^i(t)\,dt.

An increasing reparametrization preserves the value; a decreasing one reverses it. For a two-surface, the wedge of the two tangent directions plays the corresponding role. These are integrations of forms over oriented chains, not integrations of coordinate components in isolation.

The notation $\int_{\sigma(D)}$ can hide self-overlap, boundary identification, or singular parameters. When any is present, state whether the object is an embedded submanifold, an immersed chain with multiplicity, or a limiting parametrization. No physical flux interpretation is assumed here; a field theory must specify which form represents the flux and under what equations it is closed.

Independence from ambient coordinates

The definition must also be independent of the coordinates used on the target manifold. Suppose a second target chart is inserted between the form and the parametrization. The form components acquire inverse Jacobian factors, while every tangent vector obtained from the parametrization acquires the corresponding forward factor. Each pair cancels in the multilinear evaluation. What remains is the same top form on the parameter domain.

This check differs from reparametrization invariance. Changing target coordinates merely redescribes the same ambient points and form. Changing parameters replaces the map from the domain, so its determinant also tests the chosen orientation and multiplicity. Keeping these two transformations separate makes surface calculations substantially less error-prone.

For example, the area-form calculation on the disk can be performed using Cartesian target coordinates and polar parameters, or using a polar target chart away from the origin. In the second description the target top form already carries the radial Jacobian factor, while the parametrization may become the identity. Both routes give the same integral. The agreement confirms that the radial factor belongs to the form’s coordinate expression, not to a metric assumption.

Patching a manifold integral

A general compact oriented manifold need not admit one global parametrization. Choose an oriented atlas and a smooth partition of unity subordinate to it. Multiply the form by each partition function, pull each compactly supported piece into its chart, integrate there, and sum. Linearity and the change-of-variables theorem ensure that refining the atlas or partition does not alter the answer.

Overlap terms are not counted twice because the partition functions sum to one. Orientation compatibility ensures every transition determinant is positive. If an overlap reverses orientation, the signed local expressions cannot define one oriented integral until one chart orientation is reversed or the manifold is recognized as nonorientable.

This patching argument explains why the simple definition through one parametrization is not merely a special computational trick. It is the local building block of the global integral. It also shows where the standard manifold assumptions enter: partitions of unity rely on paracompact behavior, while compact support keeps every local Euclidean integral controlled.

Degenerate and repeated parametrizations

Some familiar coordinates are not diffeomorphisms on their displayed closed domains. Polar coordinates identify the two angular edges and collapse an entire angular interval at the origin. The disk integral remains valid because one can remove a narrow seam and a small central disk, integrate on a genuine chart, and then take a controlled limit. That reasoning should be stated whenever a singular parametrization is used in a proof rather than a routine convergent calculation.

A repeated parametrization is different. If a circle is traversed twice with the same orientation, its line integral is doubled; no measure-zero exception removes the multiplicity. Traversing once in each direction makes the signed contributions cancel. Thus an integral is sensitive to an oriented parametrized chain, not merely to the set of points in its image.

Exercises

Let $C$ be the unit circle with counterclockwise parametrization $\gamma(t)=(\cos t,\sin t)$ , $0\le t\le2\pi$ , and let $A=x\,dy-y\,dx$ . Compute $\int_C A$ , then repeat with $\widetilde\gamma(t)=\gamma(2\pi-t)$ .

Solution

Along $\gamma$ , $dx=-\sin t\,dt$ and $dy=\cos t\,dt$ , so

\gamma^*A=(\cos^2t+\sin^2t)dt=dt.

Therefore $\int_C A=2\pi$ . The map $t\mapsto2\pi-t$ reverses orientation, so $\widetilde\gamma^*A=-dt$ after using a consistently increasing parameter on $[0,2\pi]$ , and the integral is $-2\pi$ . The geometric trace is the same; its oriented traversal is not.

References

Guillemin, Victor, and Peter J. Haine. Differential Forms. World Scientific, 2019, §4.5, pp. 142–146, PDF.