Manifolds 23 | Differential (Definition)

Q: How is the differential $d f_p$ defined using tangent vectors and curves?

A tangent vector at $p$ is an equivalence class of curves $\gamma$ with $\gamma(0)=p$. The differential $d f_p$ takes that equivalence class and maps it to the equivalence class of the composed curve $f\circ \gamma$, which passes through $f(p)$. This produces a linear map $T_pM \to T_{f(p)}N$ and is well-defined because equivalent curves yield equivalent composed curves.

Q: What geometric meaning does $d f_p$ carry?

$d f_p$ provides a linear approximation to $f$ near $p$. Just as the derivative in multivariable calculus gives the best local linear model, the differential gives the corresponding linear map between tangent spaces. It depends on the base point $p$, which is why the notation includes the subscript.

Q: In the embedded submanifold case, how does the differential relate to directional derivatives?

When $M$ and $N$ are smooth submanifolds of $\mathbb{R}^n$, tangent vectors can be represented concretely via curves in $\mathbb{R}^n$. Applying $d f_p$ to a tangent vector corresponds to taking the directional derivative of $f$ along that vector. In coordinates, this becomes Jacobian-matrix multiplication: the Jacobian of $f$ at $p$ multiplies the derivative of the curve at the parameter value where the curve hits $p$.

Q: How does the Jacobian matrix appear in the manifold differential?

For a $C^1$ map $f: \mathbb{R}^n \to \mathbb{R}^m$ used to define the submanifold setting, the chain rule turns the derivative of $f\circ \gamma$ into $Df(p)\cdot \gamma'(0)$. Here $Df(p)$ is represented by the Jacobian matrix at $p$, and $\gamma'(0)$ encodes the tangent vector. This shows the differential generalizes the Jacobian/total derivative.

TL;DR

The differential $d f_{p}$ for a smooth map $f : M \to N$ is a linear map $T_{p} M \to T_{f (p)} N$ that depends on the base point $p$ .

Briefing Cornell Notes

Briefing

A differential for smooth maps between manifolds is built by pushing tangent vectors forward along the map—turning the familiar “derivative” idea into a coordinate-free linear approximation. For a smooth map $f : M \to N$ and a point $p \in M$ , the differential $d f_{p}$ is a linear map from the tangent space $T_{p} M$ to $T_{f (p)} N$ . Concretely, a tangent vector is represented as an equivalence class of curves $γ$ through $p$ , and $d f_{p}$ sends that class to the equivalence class of the composed curve $f \circ γ$ through $f (p)$ . This construction matters because it provides the manifold version of the “best linear approximation” that underlies calculus, now valid on curved spaces.

Getting to that point requires assembling tangent spaces across an entire manifold. At each point $p$ , there is a tangent space $T_{p} M$ , and collecting all of them produces the tangent bundle $T M$ , formed as a disjoint union of the fibers $T_{p} M$ over all points $p \in M$ . The transcript emphasizes that using a true disjoint union is important: tangent spaces at different points are distinct even if they are isomorphic as vector spaces. It also notes that the tangent bundle can be given a smooth manifold structure with dimension doubled relative to $M$ , matching the intuition that it behaves like a Cartesian product locally.

With tangent spaces in place, the differential becomes the manifold analogue of the derivative. The map $d f_{p}$ is “fixed at $p$ ”—that dependence is why the notation includes the subscript. The differential is then interpreted as the linear approximation to $f$ near $p$ , paralleling how the derivative approximates functions in ordinary calculus. In this framework, the differential also induces a global map $df$ that assigns to each point $p$ its corresponding linear map $d f_{p}$ .

The transcript then grounds the abstract definition using embedded submanifolds of $R^{n}$ . When $M$ and $N$ sit inside $R^{n}$ , the tangent spaces can be treated concretely, and the differential can be computed using ordinary derivatives. A key result emerges: the differential acting on a tangent vector corresponds to the directional derivative of $f$ along that vector. In coordinates, this directional derivative is expressed through the Jacobian matrix of $f$ at $p$ , combined via the chain rule with the derivative of the curve representing the tangent vector. The upshot is that the manifold differential generalizes the Jacobian / total derivative from multivariable calculus, while reducing to the same familiar objects in the submanifold setting.

Overall, the differential is presented as the coordinate-free mechanism that turns smooth maps between manifolds into linear maps between tangent spaces, preserving the calculus intuition of linearization—while making it work on curved domains and targets.

Cornell Notes

For a smooth map $f : M \to N$ and a point $p \in M$ , the differential $d f_{p}$ is a linear map $T_{p} M \to T_{f (p)} N$ . Tangent vectors are equivalence classes of curves $γ$ through $p$ , and $d f_{p}$ sends the class of $γ$ to the class of $f \circ γ$ . This makes $d f_{p}$ the manifold version of the derivative’s linear approximation near $p$ . For embedded submanifolds of $R^{n}$ , the construction becomes concrete: $d f_{p}$ matches the directional derivative, expressed using the Jacobian matrix and the chain rule. The differential therefore generalizes the Jacobian/total derivative from multivariable calculus to arbitrary smooth manifolds.

How is the differential $d f_{p}$ defined using tangent vectors and curves?

A tangent vector at

p

is an equivalence class of curves

γ

with

γ (0) = p

. The differential

d f_{p}

takes that equivalence class and maps it to the equivalence class of the composed curve

f \circ γ

, which passes through

f (p)

. This produces a linear map

T_{p} M \to T_{f (p)} N

and is well-defined because equivalent curves yield equivalent composed curves.

Why does the tangent bundle $T M$ use a disjoint union rather than an ordinary union?

Tangent spaces at different points are distinct fibers even when they are isomorphic. Forming

T M

as a disjoint union

⨆_{p \in M} T_{p} M

keeps

T_{p} M

and

T_{q} M

separate for

p \neq = q

. The transcript notes that using an ordinary union can break the intended identification in settings like submanifolds, so the disjoint union is the safer formal choice.

What geometric meaning does $d f_{p}$ carry?

d f_{p}

provides a linear approximation to

f

near

p

. Just as the derivative in multivariable calculus gives the best local linear model, the differential gives the corresponding linear map between tangent spaces. It depends on the base point

p

, which is why the notation includes the subscript.

In the embedded submanifold case, how does the differential relate to directional derivatives?

When

M

and

N

are smooth submanifolds of

R^{n}

, tangent vectors can be represented concretely via curves in

R^{n}

. Applying

d f_{p}

to a tangent vector corresponds to taking the directional derivative of

f

along that vector. In coordinates, this becomes Jacobian-matrix multiplication: the Jacobian of

f

p

multiplies the derivative of the curve at the parameter value where the curve hits

p

How does the Jacobian matrix appear in the manifold differential?

For a

C^{1}

map

f : R^{n} \to R^{m}

used to define the submanifold setting, the chain rule turns the derivative of

f \circ γ

into

D f (p) \cdot γ^{'} (0)

. Here

D f (p)

is represented by the Jacobian matrix at

p

, and

γ^{'} (0)

encodes the tangent vector. This shows the differential generalizes the Jacobian/total derivative.

Review Questions

How does the equivalence class of a curve $γ$ through $p$ determine a tangent vector, and how does $d f_{p}$ act on that class?
Why is the tangent bundle $T M$ constructed as a disjoint union of $T_{p} M$ over $p \in M$ , and what goes wrong with an ordinary union in some contexts?
In the submanifold-in- $R^{n}$ setting, how does the chain rule connect $d f_{p}$ to the Jacobian matrix and directional derivatives?

Key Points

1
The differential $d f_{p}$ for a smooth map $f : M \to N$ is a linear map $T_{p} M \to T_{f (p)} N$ that depends on the base point $p$ .
2
Tangent vectors are equivalence classes of curves through $p$ , and $d f_{p}$ is defined by composing those curves with $f$ .
3
Collecting all tangent spaces $T_{p} M$ across a manifold produces the tangent bundle $T M$ as a disjoint union, keeping fibers at different points separate.
4
The tangent bundle $T M$ can be given a smooth manifold structure whose dimension is twice the dimension of $M$ .
5
For embedded submanifolds of $R^{n}$ , the differential matches the directional derivative along the chosen tangent vector.
6
In coordinates, the differential reduces to Jacobian-matrix multiplication via the chain rule, generalizing the multivariable total derivative.

Highlights

The differential dfp​ is defined by pushing forward tangent vectors: a curve γ through p becomes f∘γ through f(p).

The tangent bundle TM is built as a disjoint union ⨆p∈M​Tp​M, reflecting that tangent spaces at different points are distinct fibers.

For submanifolds in Rn, dfp​ reproduces directional derivatives and is computed using the Jacobian matrix and the chain rule.

Topics

Tangent Bundle
Differential of Smooth Maps
Tangent Spaces
Directional Derivatives
Jacobian Matrix