Chapter 2(Smooth Maps)

Notation
1. Function
> codomain is that a ***real-valued function***
2. Map or Mapping
>Any type of map, such as a map between arbitrary manifolds.

Smooth Functions and Smooth Maps

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Smooth Functions on Manifolds

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Smooth Function

Definition. Smooth function $\mathbf{M}$ is a smooth $n$-manifold, $k\in {\mathbf{Z}}_{\ge 0}$ , and $f:\mathbf{M}\to {\mathbb{R}}^k$ is any function. Then $f$ is a smooth function if $\forall p\in \mathbf{M}$, there exists a smooth chart $\left(U,\varphi \right)$ for $\mathbf{M}$ whose domain contains $p$ and such that the composite function $f\circ {\varphi }^{-1}$ is a smooth on the open subset $\hat{U}=\varphi \left(U\right)\subseteq {\mathbb{R}}^n$.

If $\mathbf{M}$ is a smooth manifold with boundary, $\varphi (U)$ is now an open subset of either ${\mathbb{R}}^n$ or ${\mathbb{H}}^n$

Remarkd and Exercise 2.1 Linear AlgebraAlgebra

$C^{\infty }\left(\mathbf{M}\right)=\left\{f:\mathbf{M}\to \mathbf{R}\text{which is smooth real-valued functions}\right\}$ Property

1. $C^{\infty }\left(\mathbf{M}\right)$ is a vector space over $\mathbb{R}$.
2. Commutative Ring
3. A commutative and associative algebra over R bilinear.

Exercise 2.2

$U$ be an open submanifold of ${\mathbb{R}}^n$ with its standard smooth manifold structure. $f:U\to {\mathbb{R}}^k$ is smooth in the sense just defined $⟺$ it is a smooth in the sense of ordinary calculus. Also for open submanifold with boundary in ${\mathbb{H}}^n$

Exercise 2.3

$\mathbf{M}$ : smooth manifold with or without boundary, and $f:\mathbf{M}\to {\mathbb{R}}^k$ is a smooth function. Then $f\circ {\varphi }^{-1}:\varphi \left(U\right)\to {\mathbb{R}}^k$ is smooth for every smooth chart $\left(U,\varphi \right)$ for $\mathbf{M}$.

Proof. Since $f$ is a smooth function, $\exists$ smooth chart $\left(V,\psi \right)$ that chart domain and $f\circ {\psi }^{-1}$ is a smooth function. Also, $\mathbb{M}$ is a smooth manifold that $\psi \circ {\varphi }^{-1}$ is smooth function. Therefore, $f\circ {\psi }^{-1}\circ \psi \circ {\varphi }^{-1}=f\circ {\varphi }^{-1}$ is a smooth for every smooth chart $\left(U,\varphi \right)$ for $\mathbb{M}$.

Coordinate representation of $f$

Definition. Coordinate Representation of $f$ Given a function $f:\mathbf{M}\to {\mathbf{R}}^k$ and a chart $\left(U,\varphi \right)$ for $\mathbf{M}$, the function $\hat{f}:\varphi \left(U\right)\to {\mathbb{R}}^k$ defined by $\hat{f}(x)=f\circ {\varphi }^{-1}(x)$ is called the coordinate representation of $f$.

Property.

1. $f$ is smooth $⟺$ its coordinate representation is smooth in some smooth chart around each point. By definition
2. Smooth functions have smooth coordinate representations in every smooth chart.

Meaning In keeping with our practice of using local coordinates to identify an open subset of a manifold with an open subset of Euclidean space, in cases where it causes no confusion we often do not even observe the distinction between $\hat{f}$ and $f$ itself. 결국 표현할 수 있는 하나만 구하면 된다는 이야기임

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Smooth Maps Between Manifolds

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Partitions of Unity

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