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tutorial4.tex
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\section*{Tutorial 4 Differentiable Manifolds}
EY : 20151109 The \url{gravity-and-light.org} website, where you can download the tutorial sheets \emph{and} the full length videos for the tutorials and lectures, are no longer there. $=($
Hopefully, the YouTube video will remain: \url{https://youtu.be/FXPdKxOq1KA?list=PLFeEvEPtX_0RQ1ys-7VIsKlBWz7RX-FaL}
\exercisehead{1: True or false?} \emph{These basic questions are designed to spark discussion and as a self-test.}
Tick the correct statements, but not the incorrect ones!
\begin{enumerate}
\item[(a)] The function $f: \mathbb{R} \to \mathbb{R}$, \dots
\begin{itemize}
\item
\item
\item \dots , defined by $f(x) = |x^3|$, lies in $C^3(\mathbb{R} \to \mathbb{R})$.
\solutionhead{1a3} For $f: \mathbb{R} \to \mathbb{R}$, $f(x) = |x^3| = \begin{cases} x^3 & \text{ if } x \geq 0 \\
-x^3 & \text{ if } x < 0 \end{cases}$
\[
\begin{aligned}
& f'(x) = \begin{cases} 3x^2 & \text{ if } x \geq 0 \\
-3x^2 & \text{ if } x < 0 \end{cases} \\
& f''(x) = \begin{cases} 6x & \text{ if } x \geq 0 \\
-6x & \text{ if } x < 0 \end{cases}
\end{aligned}
\]
Thus,
\[
\boxed{ f(x) = |x^3| \in C^1(\mathbb{R}) \text{ but } f(x) \notin C^2(\mathbb{R}) \subseteq C^3(\mathbb{R}) }
\]
\item
\item
\end{itemize}
\item[(b)]
\item[(c)]
\end{enumerate}
\textbf{Short} \exercisehead{4: Undergraduate multi-dimensional analysis }
\emph{A good notation and basic results for partial differentiation}.
For a map $f: \mathbb{R}^d \to \mathbb{R}$ we denote by the map $\partial_i f: \mathbb{R}^d \to \mathbb{R}$ the partial derivative with respect to the $i$-th entry.
\questionhead{:} Given a function
\[
f: \mathbb{R}^3 \to \mathbb{R}; \, (\alpha, \beta, \delta) \mapsto f(\alpha,\beta,\delta) := \alpha^3\beta^2 + \beta^2 \delta + \delta
\]
calculate the values of the following derivatives:
\solutionhead{:}
\begin{itemize}
\item $(\partial_2f)(x,y,z) = $
\item $(\partial_1f)(\square,\circ,*) =$
\item $(\partial_1 \partial_2 f)(a,b,c) = $
\item $(\partial_3^2 f)(299,1222,0) =$
\end{itemize}
EY: 20151110
For $f(\alpha,\beta,\delta) := \alpha^3\beta^2 + \beta^2 \delta + \delta$, or $f(x,y,z) = x^3 y^2 + y^2 z + z$,
\[
\begin{aligned}
& (\partial_2 f) = 2(x^3y+yz) \\
& (\partial_1 f) = 3x^2 y^2 \\
& (\partial_1\partial_2 f) = 6x^2 y \\
& (\partial_3^2f) = 0
\end{aligned}
\]
and so
\begin{itemize}
\item $(\partial_2f)(x,y,z) = 2(x^3 y + yz) $
\item $(\partial_1f)(\square,\circ,*) = 3\square^2 \circ^2$
\item $(\partial_1 \partial_2 f)(a,b,c) = 6a^2 b$
\item $(\partial_3^2 f)(299,1222,0) = 0$
\end{itemize}
\exercisehead{5: Differentiability on a manifold}
\emph{How to deal with functions and curves in a chart}
Let $(M, \mathcal{O}, \mathcal{A})$ be a smooth $d$-dimensional manifold. Consider a chart $(U,x)$ of the atlas $\mathcal{A}$ together with a smooth curve $\gamma : \mathbb{R} \to U$ and a smooth function $f:U \to \mathbb{R}$ on the domain $U$ of the chart.
\questionhead{:} Draw a commutative diagram containing the chart domain, chart map, function, curveand the respective representatives of the function and the curve in the chart.
\solutionhead{:}
\begin{tikzpicture}[decoration=snake]
\matrix (m) [matrix of math nodes, row sep=4em, column sep=6em, minimum width=2em]
{
\mathbb{R} & U & \mathbb{R}^d \\
& \mathbb{R} & \\
};
\path[->]
(m-1-1) edge node [above] {$\gamma$} (m-1-2)
edge [bend left=40] node [auto] {$x\circ \gamma$} (m-1-3)
(m-1-3) edge [bend left=15] node [auto] {$x^{-1}$} (m-1-2)
edge node [right] {$(f\circ x^{-1})$ } (m-2-2)
(m-1-2) edge node [left] {$f$} (m-2-2)
edge node [auto] {$x$} (m-1-3);
\end{tikzpicture} \quad \quad \, \begin{tikzpicture}[decoration=snake]
\matrix (m) [matrix of math nodes, row sep=4em, column sep=6em, minimum width=2em]
{
\tau \in \mathbb{R} & p \in U & x(p) = (x\circ \gamma)(\tau) \in \mathbb{R}^d \\
& f(p) \in \mathbb{R} & \\
};
\path[|->]
(m-1-1) edge node [above] {$\gamma$} (m-1-2)
edge [bend left=40] node [auto] {$x\circ \gamma$} (m-1-3)
(m-1-3) edge [bend left=15] node [auto] {$x^{-1}$} (m-1-2)
edge node [right] {$(f\circ x^{-1})$ } (m-2-2)
(m-1-2) edge node [left] {$f$} (m-2-2)
edge node [auto] {$x$} (m-1-3);
\end{tikzpicture}
\questionhead{:} Consider, for $d=2$,
\[
(x\circ \gamma)(\lambda):= (\cos{(\lambda)}, \sin{(\lambda)} ) \text{ and } (f\circ x^{-1})((x,y)) := x^2 +y^2
\]
Using the chain rule, calculate
\[
(f\circ \gamma)'(\lambda)
\]
explicitly.
\solutionhead{:}
EY : 20151109 Indeed, the domains and codomains of this $f\gamma$ mapping makes sense, from $\mathbb{R} \to \mathbb{R}$ for
\begin{tikzpicture}[decoration=snake]
\matrix (m) [matrix of math nodes, row sep=4em, column sep=6em, minimum width=2em]
{
\mathbb{R} & U & \mathbb{R}^d \\
& \mathbb{R} & \\
};
\path[->]
(m-1-1) edge node [above] {$\gamma$} (m-1-2)
edge [bend left=40] node [auto] {$x\circ \gamma$} (m-1-3)
edge node [auto] {$f\circ \gamma$} (m-2-2)
(m-1-3) edge [bend left=15] node [auto] {$x^{-1}$} (m-1-2)
edge node [right] {$(f\circ x^{-1})$ } (m-2-2)
(m-1-2) edge node [left] {$f$} (m-2-2)
edge node [auto] {$x$} (m-1-3);
\end{tikzpicture} \quad \quad \, \begin{tikzpicture}[decoration=snake]
\matrix (m) [matrix of math nodes, row sep=4em, column sep=6em, minimum width=2em]
{
\tau \in \mathbb{R} & p \in U & x(p) = (x\circ \gamma)(\tau) \in \mathbb{R}^d \\
& f(p) \in \mathbb{R} & \\
};
\path[|->]
(m-1-1) edge node [above] {$\gamma$} (m-1-2)
edge [bend left=40] node [auto] {$x\circ \gamma$} (m-1-3)
edge node [auto] {$f\circ \gamma$} (m-2-2)
(m-1-3) edge [bend left=15] node [auto] {$x^{-1}$} (m-1-2)
edge node [right] {$(f\circ x^{-1})$ } (m-2-2)
(m-1-2) edge node [left] {$f$} (m-2-2)
edge node [auto] {$x$} (m-1-3);
\end{tikzpicture}
\[
\begin{gathered}
(f\circ \gamma)'(\lambda) = (Df)\cdot \dot{\gamma}(\lambda) = \frac{ \partial f}{ \partial x^j} \dot{\gamma}^j(\lambda) = 2x (-\sin{\lambda} ) + 2y \cos{\lambda} = 2(-\cos{\lambda} \sin{\lambda} + \sin{\lambda} \cos{\lambda} ) = 0
\end{gathered}
\]