# On the diffetential of a mapping 2

In the case ${\psi : \mathbb{R}^n \rightarrow \mathbb{R}^m}$ case, there is a linear map, which is “linear approximation” of ${\psi}$. In the manifold case, there is a similar linear map, but now it acts between tangent spaces. If ${M}$ and ${N}$ are smooth manifolds and ${\psi \colon M \rightarrow N}$ is a smooth map then for each ${m \in M}$, the map

$\displaystyle d\psi \colon T_mM \rightarrow T_{\psi(m)}N$

is defined by

$\displaystyle d\psi(v)(f) = v(f \circ \psi)$

is called the pushforward. Actually,

$\displaystyle d\psi \colon TM \rightarrow TN.$

Suppose that ${\dim{M} \ge \dim{N}}$ and ${f \colon M \rightarrow N}$ is a differentiable mapping. We have

Definition 1 The mapping ${f}$ is called a trivial fibration (differentiable) on ${N}$ if there exists a differential manifold ${F}$, is called fibre of ${f}$, and a diffeomorphism

$\displaystyle \phi \colon M \rightarrow N \times F$

such that the following diagram is commutative

# On the differential of a mapping

From on Warner’s book.

Smooth curve on manifold ${M}$:

A ${C^\infty}$ mapping ${\alpha : (a, b) \rightarrow M}$. Let ${t \in (a, b)}$, we define the tangent vector of the curve ${\alpha}$ at ${t}$ is the vector

$\displaystyle d\alpha\bigg(\dfrac{d}{dt}\bigg|_{t = 0}\bigg) \in T_{\alpha(0)}M.$

we apply the formula

$\displaystyle d\psi(v)(g) = v(g \circ \psi),$

where ${g}$ is an any function on ${M}$.

Put ${\psi = \alpha(t)}$ and ${v = \dfrac{d}{dt}\alpha(t)\big|_{t = 0}}$, the above formula implies

$\displaystyle d\alpha(\frac{d}{dt}\big|_{t=0})(f) = (\frac{d}{dt}\big|_{t=0})(f \circ \alpha) = \frac{d}{dt}(f \circ \alpha)\big|_{t=0}.$

This is directional derivative.

# Exercise of diff topo 1

BT của Munkres,

BT1, Cho $M$ là một đa tạp chiều $m$ có biên khác rỗng. Gọi $M_0 = M \times 0$$M_1 = M \times 1$ là hai mảnh của $M$. “Double” của $M$, ký hiệu là $D(M)$, là không gian tô pô tạo bởi $M_0 \cup M_1$ bằng cách gắn đồng nhất $(x,0)$$(x,1)$ với mỗi $x \in Bd(M)$ (biên của M). Chứng minh rằng $D(M)$ là một đa tạp không có biên $m$ chiều.