# Monday, September 19

:::{.remark}
Last week: tangent spaces, now working toward tangent bundles.
To give a basis for $\T_p M$, choose a smooth coordinate chart $\phi_p \in C^\infty(U, \phi(U) \subseteq \RR^n)$ about $p$ to induce an isomorphism $d\phi_p \in \Vect(\T_p U, \T_{\varphi(p)}\RR^n )$.
The latter has a basis $\ts{ \dd{}{x_i} \evalfrom{\phi(p)}  }$, so applying the inverse yields a basis $\ts{ (d\phi_p)\inv \dd{}{x_i}\evalfrom{\phi(p)} } \subseteq \T_p M$. 

A notational shift: rewrite coordinate charts as $\tv{x_1,\cdots,x_n}: U\to \phi(U) \subseteq \RR^n$ where $x_i$ are the coordinates of $\phi$ regarded as functions $U\to \RR$.
We also write the basis of $\T_p M$ as $\dd{}{x_i} \evalfrom{p}$.
Similarly, if $f: U\to \RR^n$, we write $f(x_1, x_2,\cdots,x_n)$.
If $\psi_p = (y_1,\cdots, y_n): V\to \psi(V) \subseteq \RR^n$ is another coordinate chart, then we get a different basis $\ts{\dd{}{y_i}\evalfrom{p}}$ for $\T_p M$ where $\dd{}{y_i}\evalfrom{p} \da (d\psi_p)\inv \qty{\dd{}{y_i}\evalfrom{\psi(p)} }$.

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How are these related? $\psi \circ \phi\inv: \phi(U \intersect V) \to \psi(U \intersect V)$ sends $x_i(p)$ to $y_i(p)$.
Given $v\in \T_p M$, write it as $v = \sum v_i \dd{}{x_i} \evalfrom{p}, \sum w_i \dd{}{y_i}\evalfrom{p}$ -- when are these equal?
Note $(d\phi_p)\inv = (d\phi\inv)_{\phi(p)}$ by the chain rule, and write the RHS as $(d\phi_p)\inv \qty{v_j x^j \evalfrom{\phi(p)}}$ and similarly for the LHS.
Take $(d\psi_p)$ of both sides to obtain
\[
w_i y^i \evalfrom{\psi(p)} = d(\psi \circ \phi\inv)_{\phi(p)} v_j x^j \evalfrom{p}
.\]
Now use that $d(\psi \circ \varphi\inv)$ is represented by the matrix $M \da \left[ \dd{(\psi\circ \phi\inv)_i }{x_j} \right]$, and $M = N \da \left[ \dd{y_i}{x_j} \right]$ iff $\vector w = N \vector v$.
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:::{.example title="?"}
Consider $M = \RR^2 \contains U \da \RR\cross \RR_{> 0}$.
Take $\phi: U \to \RR_{> 0} \times (0, \pi)$ sending $\vector x$ to its polar coordinates $(r, \theta)$, and set $V\da \RR^2$ and let $\psi: V\to \RR^2$ by the identity (i.e. taking Cartesian coordinates).
Note that 
\[
\psi\circ \phi\inv: \RR_{>0}\times (0, \pi) &\to \RR \times \RR_{>0} \\
(r, \theta) &\mapsto (r \cos \theta, r \sin \theta)
\]
and $N = \matt{\cos \theta}{-r \sin \theta}{\sin \theta}{r \cos \theta}$.
Thus if $v = a \dd{}{r} + b\dd{}{\theta} = c \dd{}{x} + d\dd{}{y}$, then
\[
\tv{c, d}^t = \matt{\cos \theta}{-r \sin \theta}{\sin \theta}{r \cos \theta} \tv{a, b}^t
.\]
For example, $\dd{}{r} = \cos(\theta) \dd{}{x} + \sin(\theta) \dd{}{y} = {1\over \sqrt{x^2+y^2}}\qty{x \dd{}{x} + y\dd{}{y}}$, which when applied to a function $f$ measures how it changes as a function of distance from the origin with $\theta_0$ fixed.
One can similarly compute $\dd{}{\theta} = -y\dd{}{x} + x\dd{}{y}$.
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:::{.remark}
We'll define the tangent bundle as a set as $\T M\da \ts{(p, v) \st p\in P, v\in \T_p M}$.
Later we'll topologize this and give it a smooth manifold structure.
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