数学分析笔记——An Introduction to Manifolds(2)

数学分析笔记——An Introduction to Manifolds(2)

Chapter 1 Euclidean Spaces

§2 Tangent Vectors in $\mathbb{R}^n$ as Derivations

2.1 The Directional Derivative

Elements of the tangent space $T_p(\mathbb{R}^n)$ at $p$ in $\mathbb{R}^n$ are called tangent vectors at $p$ in $\mathbb{R}^n$.

If $f$ is $C^{\infty}$ in a neighborhood of $p$ in $\mathbb{R}^n$ and $v$ is a tangent vector at $p$, the directional derivative of $f$ in the direction $v$ at $p$ is defined to be
$$
D_v f=\lim\limits_{t\to 0}\frac{f(p+tv)-f(p)}{t}=\sum\limits_{i=1}^n v^i\frac{\partial f}{\partial x^i}(p).
$$
We write
$$
D_v=\sum\left.v^i\frac{\partial}{\partial x^i}\right|_p
$$
for the map that sends a function $f$ to the number $D_v f$.

2.2 Germs of Functions

Definition. A relation on a set $S$ is a subset $R$ of $S\times S$. Given $x,y$ in $S$, we write $x\sim y$ if and only if $(x,y)\in R$. The relation $R$ is an equivalence relation if it satisfies the following three properties for all $x,y,z\in S$:

  1. (reflexivity) $x\sim x$,
  2. (symmetry) if $x\sim y$, then $y\sim x$,
  3. (transitivity) if $x\sim y$ and $y\sim z$, then $x\sim z$.

As long as two functions agree on some neighborhood of a point $p$, they will have the same directional derivatives at $p$. This suggests that we introduce an equivalence relation on the $C^\infty$ functions defined in some neighborhood of $p$. Consider the set of all pairs $(f,U)$, where $U$ is a neighborhood of $p$ and $f:U\to \mathbb{R}$ is a $C^{\infty}$ function. We say that $(f,U)$ is equivalent to $(g,V)$ if there is an open set $W \subset U\cap V$ containing $p$ such that $f =g$ when restricted to $W$. This is clearly an equivalence relation because it is reflexive, symmetric, and transitive. The equivalence class of $(f,U)$ is called the germ of $f$ at $p$. We write $C^{\infty}_p(\mathbb{R}^n)$, or simply $C^{\infty}_p$ if there is no possibility of confusion, for the set of all germs of $C^\infty$ functions on $\mathbb{R}^n$ at $p$.

Definition. An algebra over a field $K$ is a vector space $A$ over $K$ with a multiplication map
$$
\mu:A\times A\to A,
$$
usually written $\mu(a,b)=a\cdot b$, such that for all $a,b,c\in A$ and $r\in K$,

  1. (associativity) $(a\cdot b)\cdot c=a\cdot(b\cdot c)$,
  2. (distributivity) $(a+b)\cdot c=a\cdot c+b\cdot c$ and $a\cdot(b+c)=a\cdot b+a\cdot c$,
  3. (homogeneity) $r(a\cdot b)=(ra)\cdot b=a\cdot(rb).$

Equivalently, an algebra over a field $K$ is a ring $A$ (with or without multiplicative identity) that is also a vector space over $K$ such that the ring multiplication satisfies the homogeneity condition. Thus, an algebra has three operations: the addition and multiplication of a ring and the scalar multiplication of a vector space. Usually we omit the multiplication sign and write $ab$ instead of $a\cdot b$.

Definition. A map $L:V\to W$ between vector spaces over a field $K$ is called a linear map or a linear operator if for any $r\in K$ and $u,v\in V$,

  1. $L(u+v) = L(u)+L(v)$,
  2. $L(rv) = rL(v)$.

To emphasize the fact that the scalars are in the field $K$, such a map is also said to be $K$-linear.

If $A$ and $A’$ are algebras over a field $K$, then an algebra homomorphism is a linear map $L:A\to A’$ that preserves the algebra multiplication: $L(ab)=L(a)L(b)$ for all $a,b\in A$.

2.3 Derivations at a Point

For each tangent vector $v$ at a point $p$ in $\mathbb{R}^n$ , the directional derivative at $p$ gives a map of real vector spaces
$$
D_v:C_p^\infty\to\mathbb{R}.
$$
$D_v$ is $\mathbb{R}$-linear and satisfies the Leibniz rule
$$
D_v(fg)=(D_vf)g(p)+f(p)D_vg.
$$
In general, any linear map $D:C_p^\infty\to\mathbb{R}$ satisfying the Leibniz rule is called a derivation at $p$ or a point-derivation of $C_p^\infty$. Denote the set of all derivations at $p$ by $\mathcal{D}_p(\mathbb{R}^n)$. This set is in fact a real vector space.

Theorem. The linear map
$$
\begin{align}
\phi:T_p(\mathbb{R}^n)&\to\mathcal{D}_p(\mathbb{R}^n),\\
v&\mapsto D_v=\sum\left.v^i\frac{\partial}{\partial x^i}\right|_p
\end{align}
$$
is an isomorphism of vector spaces.

2.4 Vector Fields

A vector field $X$ on an open subset $U$ of $\mathbb{R}^n$ is a function that assigns to each point $p$ in $U$ a tangent vector $X_p$ in $T_p (\mathbb{R}^n)$. Since $T_p (\mathbb{R}^n)$ has basis $\{\left.\partial/\partial x^i\right|_p\}$, the vector $X_p$ is a linear combination
$$
X_p=\sum a^i(p)\left.\frac{\partial}{\partial x^i}\right|_p,\quad p\in U,\quad a^i(p)\in\mathbb{R}.
$$
Omitting $p$, we may write $X=\sum a^i \partial/\partial x^i$, where the $a^i$ are now functions on $U$. We say that the vector field $X$ is $C^\infty$ on $U$ if the coefficient functions $a^i$ are all $C^\infty$ on $U$.

The ring of $C^\infty$ functions on an open set $U$ is commonly denoted by $C^\infty(U)$ or $\mathcal{F}(U)$. Multiplication of vector fields by functions on $U$ is defined pointwise:
$$
(fX)_p=f(p)X_p,\quad p\in U.
$$
Clearly, if $X=\sum a^i \partial/\partial x^i$ is a $C^\infty$ vector field and $f$ is a $C^\infty$ function on $U$, then $fX=\sum(fa^i)\partial/\partial x^i$ is a $C^\infty$ vector field on $U$. Thus, the set of all $C^\infty$ vector fields on $U$, denoted by $\mathfrak{X}(U)$, is not only a vector space over $\mathbb{R}$, but also a module over the ring $C^\infty(U)$. We recall the definition of a module.

Definition. If $R$ is a commutative ring with identity, then a (left) $R$-module is an abelian group $A$ with a scalar multiplication map
$$
\mu:R\times A\to A,
$$
usually written $\mu(r,a)=ra$, such that for all $r,s\in R$ and $a,b\in A$,

  1. (associativity) $(rs)a=r(sa)$,
  2. (identity) if $1$ is the multiplicative identity in $R$, then $1a=a$,
  3. (distributivity) $(r+s)a = ra+sa, r(a+b) = ra+rb$.

If $R$ is a field, then an $R$-module is precisely a vector space over $R$. In this sense, a module generalizes a vector space by allowing scalars in a ring rather than a field.

Definition. Let $A$ and $A’$ be $R$-modules. An $R$-module homomorphism from $A$ to $A’$ is a map $f:A\to A’$ that preserves both addition and scalar multiplication: for all $a,b\in A$ and $r\in R$,

  1. $f(a+b) = f(a)+f(b)$,
  2. $f(ra) = rf(a)$.

2.5 Vector Fields as Derivations

If $X$ is a $C^\infty$ vector field on an open subset $U$ of $\mathbb{R}^n$ and $f$ is a $C^\infty$ function on $U$, we
define a new function $Xf$ on $U$ by
$$
(Xf)(p)=X_pf\quad\text{for any }p\in U.
$$
Writing $X=\sum a^i\partial/\partial x^i$, we get
$$
(Xf)(p)=\sum a^i(p)\frac{\partial f}{\partial x^i}(p),
$$
or
$$
Xf=\sum a^i\frac{\partial f}{\partial x^i},
$$
which shows that $Xf$ is a $C^{\infty}$ function on $U$. Thus, a $C^\infty$ vector field $X$ gives rise to an $\mathbb{R}$-linear map
$$
\begin{align}
C^\infty(U)&\to C^\infty(U)\\
f&\mapsto Xf.
\end{align}
$$
Proposition (Leibniz rule for a vector field). If $X$ is a $C^{\infty}$ vector field and $f$ and $g$ are $C^\infty$ functions on an open subset $U$ of $\mathbb{R}^n$, then $X(fg)$ satisfies the product rule (Leibniz rule):
$$
X(fg)=(Xf)g+fXg.
$$
Definition. If $A$ is an algebra over a field $K$, a derivation of $A$ is a $K$-linear map $D:A\to A$ such that
$$
D(ab)=(Da)b+aDb,\quad\text{for all }a,b\in A.
$$
The set of all derivations of $A$ is closed under addition and scalar multiplication and forms a vector space, denoted by $\text{Der}(A)$. As noted above, a $C^\infty$ vector field on an open set $U$ gives rise to a derivation of the algebra $C^\infty(U)$. We therefore have a map
$$
\begin{align}
\varphi:\mathfrak{X}(U)&\to\text{Der}(C^{\infty}(U)),\\
X&\mapsto(f\mapsto Xf).
\end{align}
$$
Just as the tangent vectors at a point $p$ can be identified with the point-derivations of $C^\infty_p$, so the vector fields on an open set $U$ can be identified with the derivations of the algebra $C^\infty(U)$; i.e., the map $\varphi$ is an isomorphism of vector spaces.

Note that a derivation at $p$ is not a derivation of the algebra $C^\infty_p$. A derivation at $p$ is a map from $C^\infty_p$ to $\mathbb{R}$, while a derivation of the algebra $C^\infty_p$ is a map from $C^\infty_p$ to $C^\infty_p.$

 

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