Multivariable Calculus 22 | Local Diffeomorphisms

TL;DR

A global C^k diffeomorphism requires bijectivity between whole open sets, with both the map and its inverse C^k.

Briefing Cornell Notes

Briefing

Local diffeomorphisms formalize what coordinate changes do: they behave like smooth, invertible maps once you zoom in far enough, even when they fail to be globally one-to-one. The key takeaway is that a map can be a diffeomorphism on small neighborhoods—so long as it has a smooth inverse there—without being a diffeomorphism on the entire domain.

The discussion begins by contrasting global diffeomorphisms with weaker, local versions. A global C^k diffeomorphism requires a one-to-one correspondence between two open sets U and V in R^n, with both the map and its inverse continuously differentiable up to order k. But many transformations don’t stay injective or bijective everywhere. The remedy is to “zoom in”: restrict attention to smaller open sets U~ and V~ so the map becomes a C^k diffeomorphism locally. This shift matters because it matches how coordinate systems are actually used—locally—when doing analysis and computations.

Polar coordinates in two dimensions provide the central example. The map Φ takes (r, φ) to (r cos φ, r sin φ), with r restricted to positive real numbers so the origin is excluded. The Jacobian determinant is computed from the derivatives of cosine and sine, yielding det(DΦ)=r. Since r>0, the determinant never vanishes on the domain, so the map has the right local non-degeneracy for a diffeomorphism. Yet it fails globally because Φ is not injective: the angle φ can wrap around and represent the same point after multiple turns. In geometric terms, the map folds the plane repeatedly, so no single global inverse exists.

The fix is not to force a global inverse, but to restrict the domain and codomain to regions where the folding doesn’t occur. By limiting to a small “rectangle” in (r, φ)—small enough that φ doesn’t complete a full turn—the image becomes a corresponding local patch of the plane where the map is both injective and surjective onto that patch. On those neighborhoods, the polar coordinate map behaves like a genuine diffeomorphism.

From this motivation, the video gives a precise definition of a local C^k diffeomorphism. A map f:U→V is a local C^k diffeomorphism at a point x∈U if there exist open neighborhoods U~ containing x and V~ such that the restricted map f|_{U~}:U~→V~ is a C^k diffeomorphism in the usual global sense (with a C^k inverse). If this property holds for every point x in U, the map is simply called a local C^k diffeomorphism.

Finally, the same pattern appears in other coordinate systems: polar coordinates in three dimensions and cylindrical coordinates also work as local C^1 diffeomorphisms, even though they may not be globally diffeomorphic. The closing point points ahead to a criterion for checking local diffeomorphisms: the determinant of the Jacobian matrix, tied to the inverse function theorem, which will be developed next.

Cornell Notes

Local diffeomorphisms capture the idea behind coordinate changes: smooth maps that become invertible with a smooth inverse once you zoom in. A global C^k diffeomorphism needs a one-to-one correspondence between whole open sets, but many coordinate maps fail globally due to non-injectivity (like angle wrapping). The remedy is localizing: around each point x, find neighborhoods U~ and V~ so the restriction f|_{U~}:U~→V~ is a C^k diffeomorphism. Polar coordinates illustrate this: det(DΦ)=r is never zero for r>0, but the map is not globally injective because φ can turn multiple times. Restricting φ to a small range restores injectivity locally, making the map a local diffeomorphism.

Why does polar coordinates fail to be a global diffeomorphism in 2D, even though its Jacobian determinant never vanishes?

The Jacobian determinant for Φ(r,φ)=(r cos φ, r sin φ) is det(DΦ)=r, which is positive when r>0, so the map is locally non-degenerate. But Φ is not injective globally because φ can represent the same direction after multiple full rotations (angle wrapping). Without injectivity, there cannot be a global inverse map, so the global diffeomorphism requirement fails.

How does restricting the domain fix polar coordinates?

Instead of trying to invert Φ everywhere, restrict to smaller neighborhoods in (r,φ) where φ does not complete a full turn. On such a small region, the image is a corresponding patch of the plane where each point has exactly one preimage in that neighborhood. With det(DΦ)=r≠0 still holding, the restricted map becomes bijective onto its image and admits a smooth inverse there—so it is a diffeomorphism locally.

What is the formal definition of a local C^k diffeomorphism at a point?

For f:U→V, f is a local C^k diffeomorphism at x∈U if there exist open sets U~⊂U containing x and V~⊂V such that the restricted map f|_{U~}:U~→V~ is a C^k diffeomorphism in the usual sense. That means f|_{U~} is C^k, bijective between U~ and V~, and its inverse (from V~ back to U~) is also C^k.

What changes when the definition is required to hold for every point in U?

If the local C^k diffeomorphism property holds at every x∈U—meaning for each point one can find suitable neighborhoods U~ and V~—then f is called a local C^k diffeomorphism (without specifying a single point). The neighborhoods may differ from point to point.

Which coordinate systems are highlighted as local diffeomorphisms, and what limitation remains?

Polar coordinates in three dimensions and cylindrical coordinates are described as local C^1 diffeomorphisms. The limitation is that these coordinate maps may not be globally diffeomorphic; they work locally because the Jacobian non-degeneracy and injectivity can be ensured on sufficiently small neighborhoods.

What criterion is foreshadowed for checking local diffeomorphisms?

The determinant of the Jacobian matrix is identified as the key criterion, connected to the inverse function theorem. Non-vanishing of the Jacobian determinant supports local invertibility and the existence of a smooth inverse on neighborhoods.

Review Questions

In polar coordinates with r>0, what role does det(DΦ)=r play, and why doesn’t it guarantee a global diffeomorphism?
State the difference between a global C^k diffeomorphism and a local C^k diffeomorphism at a point.
How would you choose neighborhoods in the (r,φ) parameter space to make the polar coordinate map injective locally?

Key Points

1
A global C^k diffeomorphism requires bijectivity between whole open sets, with both the map and its inverse C^k.
2
Local diffeomorphisms replace global bijectivity with bijectivity on smaller neighborhoods around each point.
3
Polar coordinates in 2D have Jacobian determinant det(DΦ)=r, which is never zero when r>0, supporting local invertibility.
4
Polar coordinates still fail globally because the angle variable φ makes the map non-injective after full rotations.
5
Restricting φ to a small range (so no full turn occurs) restores injectivity locally and yields a local diffeomorphism.
6
Polar coordinates in 3D and cylindrical coordinates behave similarly: they are local C^1 diffeomorphisms even if not globally diffeomorphic.
7
The inverse function theorem and the Jacobian determinant provide the practical criterion for verifying local diffeomorphisms.

Highlights

The Jacobian determinant for the 2D polar coordinate map is det(DΦ)=r, so it never vanishes on the domain r>0.

Global failure comes from non-injectivity (angle wrapping), not from a zero Jacobian.

A local diffeomorphism is defined by finding neighborhoods U~ and V~ where the restricted map has a C^k inverse.

Polar coordinates become diffeomorphisms once φ is restricted so the map doesn’t fold the plane within that neighborhood.

Cylindrical coordinates and 3D polar coordinates are treated as local C^1 diffeomorphisms, matching how coordinate systems are used in practice.

Topics

Local Diffeomorphisms
Jacobian Determinant
Inverse Function Theorem
Coordinate Systems
Polar Coordinates