Manifolds 24 | Differential in Local Charts [dark version]

TL;DR

A smooth map F: M → N is smooth exactly when its coordinate representation F~ = K ∘ F ∘ H^{-1} is smooth as an ordinary map R^n → R^m.

Briefing Cornell Notes

Briefing

The differential on a manifold can be computed in local coordinate charts using the same machinery as multivariable calculus: Jacobian matrices and the chain rule. That connection matters because it turns an abstract definition of “differential” in terms of tangent vectors and curves into a concrete, calculable object—so manifold calculus genuinely generalizes the familiar Euclidean setting.

Start with smooth manifolds M and N and a smooth map F: M → N. Pick a point P ∈ M. By the manifold definition, there are local charts around P: an open neighborhood of P is homeomorphic to an open subset of R^n via a homeomorphism H, and a corresponding neighborhood on the target side is charted by a homeomorphism K into R^m. Smoothness of F is checked in these coordinates: F is smooth exactly when the coordinate representation F~ = K ∘ F ∘ H^{-1} is smooth as an ordinary map from R^n to R^m.

To relate differentials, choose an abstract tangent vector v ∈ T_P M. The differential D F at P acts on v by using the curve-based definition: v can be represented by a curve γ through P, and D F sends v to the tangent vector represented by F ∘ γ. In coordinates, this becomes concrete. After applying K, the abstract tangent vector corresponds to an ordinary derivative in R^n, so the action of D F can be expressed through the ordinary differential of the coordinate map F~.

The key step is to apply the multivariable chain rule to the coordinate expression. The differential of F~ at the point H(P) is represented by the Jacobian matrix J_{F~}(H(P)). When D F is written in local charts, the Jacobian matrix multiplies the differential of the chart map H at the curve parameter value (the transcript uses the parameter value 0). After that multiplication, the result is translated back from concrete tangent vectors in R^m to the abstract tangent space T_{F(P)} N using the chart-based identification.

In compact form, the differential in local charts takes the structure D F = K^{-1} ∘ J_{F~} ∘ D H, with the understanding that the Jacobian is evaluated at the appropriate coordinate point (coming from H(P)) and D H is evaluated at the corresponding parameter point. The upshot is practical: the “strange” manifold definition of differential via abstract tangent vectors is consistent with the standard Euclidean formula. Once that bridge is established, subsequent work can safely continue using the manifold-style differential while relying on Jacobians and the chain rule to compute it.

Cornell Notes

A smooth map between manifolds becomes an ordinary smooth map in local coordinates. Using charts H: M → R^n and K: N → R^m, the coordinate version F~ = K ∘ F ∘ H^{-1} has an ordinary differential represented by its Jacobian matrix J_{F~}. For an abstract tangent vector v ∈ T_P M represented by a curve γ through P, the manifold differential D F acts by pushing γ forward through F. In charts, this action becomes the chain rule: J_{F~}(H(P)) multiplies the coordinate differential D H at P, and the result is converted back to an abstract tangent vector via K^{-1}. This shows manifold differentials generalize multivariable calculus without changing the computational core.

How do local charts turn a manifold map into an ordinary multivariable map?

Pick P ∈ M and use a chart H that identifies a neighborhood of P with an open set in R^n. On the target side, use a chart K identifying a neighborhood of F(P) with an open set in R^m. Then the coordinate representation is F~ = K ∘ F ∘ H^{-1}, an ordinary map from R^n to R^m. Smoothness of F is equivalent to smoothness of F~.

Why does the differential of a manifold map reduce to the chain rule in coordinates?

An abstract tangent vector v ∈ T_P M can be represented by a curve γ with γ(0)=P. By the curve-based definition, D F(v) corresponds to the tangent vector represented by F ∘ γ. After translating to coordinates using H and K, pushing γ through F becomes pushing its coordinate version through F~, so the ordinary multivariable chain rule applies to F~.

Where does the Jacobian matrix enter the computation?

The differential of the coordinate map F~: R^n → R^m at the point H(P) is represented by the Jacobian matrix J_{F~}(H(P)). In the local-chart expression for D F, this Jacobian multiplies the coordinate differential coming from D H (evaluated at the parameter value corresponding to P, i.e., the transcript uses 0).

What role does the chart map H play in the final formula?

D H converts the abstract tangent vector at P into a concrete tangent vector in R^n coordinates. In the local-chart expression, D H is the factor that feeds into the Jacobian J_{F~}. After multiplication, the result is translated back to the abstract tangent space on the N side.

How is the result converted back from Euclidean tangent vectors to abstract tangent vectors?

After applying the Jacobian to the coordinate differential, the computation returns to the manifold’s tangent space by using the inverse chart identification on the target side, represented in the transcript by K^{-1}. This mirrors the earlier identification between abstract tangent vectors and their concrete coordinate derivatives.

Review Questions

Given charts H and K, write the coordinate form of a manifold map F and state the smoothness criterion in terms of that coordinate form.
Explain how a tangent vector represented by a curve γ through P leads to the Jacobian-based coordinate formula for D F.
In local charts, what is the order of operations involving D H and the Jacobian of F~?

Key Points

1
A smooth map F: M → N is smooth exactly when its coordinate representation F~ = K ∘ F ∘ H^{-1} is smooth as an ordinary map R^n → R^m.
2
Manifold differentials can be computed using tangent vectors represented by curves through the base point.
3
In local coordinates, the differential of F~ is represented by the Jacobian matrix J_{F~}(H(P)).
4
The chain rule governs how D F factors through the chart differential D H and the Jacobian.
5
After applying the Jacobian in coordinates, the result is translated back to the abstract tangent space using the inverse chart map K^{-1}.
6
The manifold definition of differential via abstract tangent vectors matches the Euclidean Jacobian-based computation in charts.

Highlights

The differential on a manifold becomes a Jacobian-matrix computation once expressed in local charts.

Tangent vectors defined abstractly via curves through P translate into ordinary derivatives in R^n coordinates.

The local-chart formula for D F has the same chain-rule structure as multivariable calculus: chart differential → Jacobian → chart inverse.

The abstract tangent-vector framework is validated by recovering the standard Euclidean differential in coordinates.

Topics

Manifold Differentials
Local Charts
Tangent Vectors
Jacobian Matrix
Chain Rule