---
tag: 565B0Y76XE
kind: definition
name: Differential of a smooth map
msc: [58A05]
defines: 6P9PXQDSEN
uses: [F45VQ47GPV, WY6K3P4YCG, 0V33BNKZ9S, 047NDTEPCV]
sources:
  - url: https://en.wikipedia.org/wiki/Pushforward_(differential)
---

## Definition

Let $M$ and $N$ be [[smooth manifolds|F45VQ47GPV]], let $F \colon M \to N$ be a [[smooth|0V33BNKZ9S]] map, and let $p \in M$.
Tangent vectors are taken as derivations, as in the [[tangent bundle|WY6K3P4YCG]].
The *differential* of $F$ at $p$ is the linear map
$$dF_p \colon T_pM \to T_{F(p)}N, \qquad (dF_p\, v)(f) = v(f \circ F)$$
for $v \in T_pM$ and $f \in C^\infty(N)$.
The maps $dF_p$ for all $p$ together form the smooth map $dF \colon TM \to TN$, $(p, v) \mapsto (F(p), dF_p v)$.

## Remarks

In charts $(x^1, \ldots, x^m)$ near $p$ and $(y^1, \ldots, y^n)$ near $F(p)$, with $F^j = y^j \circ F$,
$$dF_p\left(\left.\frac{\partial}{\partial x^i}\right|_p\right) = \sum_{j=1}^{n} \frac{\partial F^j}{\partial x^i}(p) \left.\frac{\partial}{\partial y^j}\right|_{F(p)},$$
so the matrix of $dF_p$ is the [[Jacobian matrix|047NDTEPCV]] of $F$ in these coordinates.

Example: for $F \colon \R \to \R^2$, $F(t) = (\cos t, \sin t)$, one gets $dF_t(d/dt) = -\sin t \, \partial_x + \cos t \, \partial_y$.
Check: for $f \in C^\infty(\R^2)$, $(d/dt)(f(\cos t, \sin t)) = -\sin t \, f_x + \cos t \, f_y$ by the chain rule.

The formula $v \mapsto v(f \circ F)$ needs $F$ to be smooth: if $F$ is only continuous, $f \circ F$ need not be smooth and $v(f \circ F)$ is not defined.

The chain rule reads $d(G \circ F)_p = dG_{F(p)} \circ dF_p$.
Other notations are $F_*$, $DF$, $TF$ and $F'$.

## Explanations

### L1 human

The differential of a smooth map at a point is the linear map that sends each velocity vector at the point to the velocity vector of the image curve.