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On this page:

• Introduction
• Smooth Functions on Euclidean Spaces
• Tangent Vectors in $\RR^n$
  • Directional Derivative
  • Germs
• Point Derivations
• Worked Exercises

Introduction

This page is my notes is from my working through the first chapter of Loring Tu’s Introduction to Manifolds.

Smooth Functions on Euclidean Spaces

[definition] [deftitle]Definition %counter% (Smooth $U\subseteq \RR^n\to \RR$)[/deftitle]

Let $U\subseteq \RR^n$ be open. We say that a real-valued function $f:U \to \RR$ is $C^k$ at $p\in U$ provided that

of all order $0 \le j \le k$ exist and are continuous at $p$. We say that $f$ is $C^\infty$ or smooth at $p$ provided that it is $C^k$ for every $k \ge 0$. [/definition]

[definition] [deftitle]Definition %counter% (Smooth $U\subseteq \RR^n\to \RR^m$)[/deftitle]

Let $U\subseteq \RR^n$ be open. We say that $f:U \to \RR^m$ is $C^k$ at $p\in U$ provided that all of its component functions $f^1, \ldots, f^m$ are $C^k$ at $p$. Furthermore, it is $C^k$ on $U$ if it is $C^k$ at all points of $U$ and we define $C^\infty$ analogously. [/definition]

[example] [extitle]Example %counter% ($C^k$ maps)[/extitle]

Via the fundamental theorem of calculus, it is straightforward to construct $C^k$ functions that are not $C^{k+1}$. To do so, we just integrate a $C^{k-1}$ function $g$:

[/example]

[definition] [deftitle]Definition %counter% (Analytic)[/deftitle]

Let $f:U \to \RR$ is real-analytic at $p \in U$ if it can be represented by its Taylor series in some neighborhood of $p$. [/definition]

[example] [extitle]Example %counter% ($C^\infty$ does NOT mean analytic)[/extitle]

Let $f:\RR \to \RR$ be defined by

It is fairly clear that $f^{(k)}(x) = P_k(x^{-1})e^{-1/x^2}$ for $x > 0$. Furthermore,

This implies that the Taylor series of $f$ is just the zero map. [/example]

Tangent Vectors in $\RR^n$

Directional Derivative

[definition] [deftitle]Definition %counter% ($T_p\RR^n$)[/deftitle]

Let $p\in \RR$. We define the tangent space $T_p\RR^n$ of $\RR^n$ at $p$ as $T_\RR^n \coloneqq \RR^n$. The elements of $T_p\RR^n$ are called tangent vectors. [/definition]

The idea with the notation above is that we can visualize the tangent and translate it to the origin to get a linear space $\RR^n$.

[definition] [deftitle]Definition %counter% (Directional derivative)[/deftitle]

Let $p \in \RR^n$ and $v$ be a tangent vector at $p$. Furthermore, let $U$ be a neighborhood of $p$ and $f:U \to \RR$ be $C^k$ on $U$. The directional derivative of $f$ in the direction of $v$ at $p$ is defined as

[/definition]

By the chain rule, we know that

Later on, we will work with the operator

We will especially be interested in the mapping $v\mapsto D_v$.

Germs

Fix some $p\in \RR^n$. We define an relation on $C^\infty$ maps of the form $f:U \to \RR^n$ where $U$ is a neighborhood of $p$ as follows: Given another $C^\infty$ map $g:V \to \RR^n$ where $V$ is a neighborhood of $p$, we say that $f \sim g$ if and only if there is an open set $W \subseteq U \cap V$ containing $p$ such that $f|_W = g|_W$. This map is an equivalence relation:

• (Reflexivity) $f\sim f$ as $U \subseteq U \cap U$.
• (Symmetric) $f\sim g$ implies $g\sim f$ as $W \subseteq U \cap V = V \cap U$.
• (Transitivity) $f\sim g$ and $g \sim h$ implies $f\sim h$ as $f\sim g$ implies the existence of open $W_1 \subseteq U\cap V$ containing $p$ such that $f|_{W_1} = g|_{W_1}$ and $g\sim h$ implies the existence of open $W_2 \subseteq U \cap V$ containing $p$ such that $f|_{W_2} = g|_{W_2}$. Setting $W = W_1 \cap W_2$, it follows that $f|_W = h|_W$ and $W \subseteq U\cap V$.

[definition] [deftitle]Definition %counter% (Germs and $C_p^\infty$)[/deftitle]

The equivalence class of $f$ at $p$ is called the germ of $f$ at $p$. The set of all germs of $C^\infty$ functions on $\RR^n$ at $p$ is denoted $C_p^\infty$. [/definition]

[example] [extitle]Example %counter%[/extitle]

The maps $f:(\RR \setminus \{1\}) \to \RR$ and $g:(-1, 1) \to \RR$ given by

represent the same germ at any point $p$ in the open $(-1, 1)$. [/example]

[theorem] [thmtitle]Proposition %counter%[/thmtitle]

$C_p^\infty$ has a natural $\RR$-algebra structure. [/theorem]

[proof] (Addition) Let $[f:U \to \RR]_\sim, [g:V \to \RR]_\sim \in C_p^\infty$ and let $W \coloneqq U \cap V$ such that $f$ and $g$ agree on $W$. Then we define

We claim that this operation is well-defined. Suppose that $[f:U\to \RR]_\sim = [f_0:U_0\to\RR]_\sim$ and $[g:V \to \RR]_\sim = [g_0:V_0\to \RR]_\sim$. So there is $W_1 \subseteq U \cap U_0$ and $W_2 \subseteq V \cap V_0$ both containing $p$ such that

Now let $W_0 = W_1 \cap W_2$. Then

so it follows that

Thus,

(Multiplication) Let $[f:U \to \RR]_\sim, [g:V \to \RR]_\sim \in C_p^\infty$ and let $W\coloneqq U \cap V$. Then we define

The well-definedness is the same argument as above.

(Scalar multiplication) Let $[f:U \to \RR]_\sim$ and let $r \in \RR$. Then we define

These operations all define an $\RR$-algebra (not necessarily unital). [/proof]

Notice that the algebra $C_p^\infty$ is given by taking a direct limit over the neighborhoods of $p$.

Point Derivations

Worked Exercises

[example] [extitle]Tu Problem 1.1[/extitle]

Let $g:\RR \to \RR$ be defined by $g(x) = \frac{3}{4}x^{4/3}$. Show that $h:\RR \to \RR$ given by $h(x) = \int_0^x g(t) \dd{t}$ is $C^2$ but not $C^3$ at $x = 0$. [/example]

[proof] By the fundamental theorem of calculus, $h’(x) = g(x)$. Thus, $h’$ is $C^1$ and so $h$ is $C^2$. [/proof]

[example] [extitle]Tu Problem 1.3[/extitle]

1. Show that the function $f:(-\pi/2, \pi/2) \to \RR$ given by $f(x) = \tan x$ is a diffeomorphism.
2. Let $a,b \in \RR$ with $a < b$. Find a linear function $h:(a, b) \to (-1, 1)$. In particular, this shows that all (bounded) open intervals are diffeomorphic.
3. The map $\exp:\RR \to (0, \infty)$ is a diffeomorphism. Use it to show that for any real numbers $a$ and $b$, the intervals $\RR$, $(a, \infty)$, and $(-\infty, b)$ are diffeomorphic. [/example]

[proof] (1) The function $\tan x$ is analytic, so $f$ is certainly $C^\infty$. Bijectivity follows from the fact that $\tan x$ is invertible on the given interval. Furthermore, $f^{-1}(x) = \arctan x$ and this is also analytic.

(2) The map $h:(a, b) \to (-1, 1)$ given by

So we have $h(a) = -1$ and $h(b) = 1$.

(3) Note that $a + e^x$ and $b - e^x$ give the desired diffeomorphisms. [/proof]

[example] [extitle]Tu Problem 1.4[/extitle]

Show that the map

is a diffeomorphism. [/example]

[proof] Each component function is analytic and have an inverse which consists of taking an inverse tangent in each component (also analytic). [/proof]