====== 2020-02-03, Monday ======

=====  Curvilinear coordinate =====

** Definition(Curviliear coordinates) **
Let $U$ be an open set in $\R^n$. A curvilinear coordinate on $U$ is a smooth function $f=(f_1,\cdots,f_n): U \to \R^n$, such that 
   * $f$ is a **bijection** between $U$ and its image $f(U)$
   * and the inverse function $f^{-1}: f(U) \to U$ is also smooth. 

** Notation ** We reserve the notation $(x_1, \cdots, x_n)$ to be the standard Cartesian coordinate on $\R^n$. We use notation of a pair $(U, (u_1, \cdots, u_n))$, or $(U, (f_1, \cdots, f_n))$ for a coordinate on $U \subset \R^n$. 

===== Tangent Vector  =====
$$\gdef\b{\mathbf}$$ 
$$\gdef\d{\partial}$$

Consider the n-dimensional Euclidean space $\R^n$ with basis vectors $\b e_1, \cdots, \b e_n$. A vector 
$\b v = v^1 \b e_1 + \cdots + v^n \b e_n$ has two possible meansings
  - it can represent a location in the space $\R^n$. You cannot add two locations (can you add New York to San Francisco?)
  - it can represent a velocity vector (an arrow with direction and length). 

In order to represent both the position and the velocity ((the use of the terminology 'velocity' is not standard in math.)), we need consider the notion of a tangent vector on $\R^n$. 

** Definition (Tangent vector) **
A tangent vector on $U \subset \R^n$ is a pair $(\b a, \b v)$ representing the location and velocity of a particle, where $\b a \in U$ represent the position, and $\b v \in \R^n$ represent the velocity. We denote the set of tangent vectors over a point $p \in U$ as $T_p U$. It is an $n$-dimensional vector space.

Warning: Only tangent vectors standing over the same position can be added or subtracted. 

** Definition (Vector field) **
A vector field on $U \subset \R^n$ is an assignment of tangent vectors $\b v$ to each point $\b a \in U$, such that $\b v$ varies smoothly with respect to $\b a$. 

==== Example: Coordinate Vector Field ====
Let $(U, (u_1, \cdots, u_n))$ be a coordinate system on $U$. Let $p \in U$ be a point, and choose an $i \in \{1, \cdots, n\}$. We will define a tangent vector $\frac{\d}{\d u_i} \vert_p$ at $p$ "physically" as follows. Consider the motion of a particle on $U$, describe by the following curve $\gamma: (-\epsilon, +\epsilon) \to U$, such that $\gamma(0) = p$, and for $t \in (-\epsilon, +\epsilon)$
$$ u_j(\gamma(t)) = \begin{cases} u_j(\gamma(0)) & j \neq i \cr u_j(\gamma(0))+t & j = i \end{cases} $$. 
Then, we define $\frac{\d}{\d u_i} \vert_p$ to be the velocity of the particle at the moment $t=0$. 

As we vary $p \in U$, the tangent vectors $\frac{\d}{\d u_i} \vert_p$ forms a vector field, denoted as $\frac{\d}{\d u_i}$ or $\d_{u_i}$. This is called a **coordinate vector field**. 

Without spelling out all the details, I will simply say that $\d_{u_i}$ generate a flow (or motion) of $U$, that moves each point on $U$ by keeping all the $u_j$ coordinates fixed and only increasing the $u_i$ coordinate "at unit speed". Imagine $\d_x$ on $\R^3$ is moving everyone towards the positive $x$ axis. 

===== Cotangent Vector =====
Recall previously, at every point $p \in U$, we have the tangent vector space $T_p U$. We can consider the dual space there and we get $T^*_p U$. An element in $T_p^* U$ is called a cotangent vector. 

Let $f$ be a smooth function on $U$. $p \in U$ a point. We will define $df(p) \in T_p^*U$ a cotangent vector at $p$. By definition, we need to specify for each tangent vector $v \in T_p U$, what is the value $df(p)(v)$. This is the directional derivative of $f$ at $p$ in the direction $v$:
$$ df(p)(v) := \sum_{i=1}^n v^i \frac{\d f}{\d x_i}\vert_p $$

We can define $df(p)(v)$ without using coordinate. Let $\gamma: (-\epsilon, +\epsilon) \to U$ be a curve, such that $\gamma(0)=p$ and $\dot \gamma(0)=v$, then 
$$ df(p)(v) = \frac{d f(\gamma(t))}{d t} \vert_{t=0}.$$

** Definition (Differential 1-form) ** A differential one-form is a assignment from $p \in U$ to elements in $T_p^*U$, that varies smoothly with $p$. 

$df$ is a differential one-form. Since the coordinates $x_1, \cdots, x_n$ are also function on $U$, we also have $dx_1, \cdots, dx_n$ as differential one-forms. 

** Lemma ** Let $u_1, \cdots, u_n$ be a coordinate on $U$. Then for each point $p$, $du_1(p), \cdots, du_n(p)$ is a basis of the cotangent vectors $T_p^* U$.

Since $\{ d u_i(p) \}$ is a basis on $T_p^*U$, we can decompose the element $df(p)$, it turns out the decomposition is as following
$$ df(p) = \frac{\d f}{\d u_1}(p) d u_1 + \cdots + \frac{ \d f}{\d u_n} (p) d u_n, $$
where the partial derivatives 
$$ \frac{\d f}{\d u_i}(p) = df(p) (\frac{\d }{\d u_i} ). $$
If one view $f$ as a function on the curvilinear coordinates $u_1, \cdots, u_n$, then these are indeed partial derivatives.