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\begin{document}

\input{213-Lecture-Macros}		%% 	macros, settings, et.c


\title{Math 213 - Surface Integrals}			%%	title page
\author{Peter A. Perry}
\institute{University of Kentucky}
\date{November 25, 2019}
\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	
%	Due Date Reminders
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Reminders}

\begin{itemize}

\item	Homework D2 is due \textbf{tonight}
\item Thanksgiving is coming!

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%	Unit Outline
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Unit IV: Vector Calculus}
\begin{center}
\begin{itemize}
\item[{}]			Fundamental Theorem for Line Integrals
\item[{}]			Green's Theorem
\item[{}]			Curl and Divergence
\item[{}]			Parametric Surfaces and their Areas
\begingroup
\color{red}
\item[{}]			Surface Integrals
\endgroup
\item[{}]			Stokes' Theorem, I
\item[{}]			Stokes' Theorem, II 
\item[{}]			The Divergence Theorem
\item[{}]			
\item[{}]			Review
\item[{}]			Review
\item[{}]			Review
\end{itemize}
\end{center}
\end{frame}

\section{Learning Goals}

\begin{frame}
\frametitle{Goals of the Day}
This lecture is about parametric surfaces. You'll learn:

\bigskip

\begin{itemize}
\item	How to integrate a scalar function over a parameterized surface
\item	What an \emph{oriented surface} is and how to compute its \emph{unit normal}
\item	How to integrate a vector field over a parameterized surface
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Sneak Preview - Scalar Surface Integrals}
\adjustbox{valign=t}
{
\begin{minipage}{0.45\textwidth}
\footnotesize
\begin{center}
\textbf{Scalar Line Integrals}
\end{center}

\medskip

\scriptsize
If $C$ is parameterized by
$$\bfr(t) = \langle x(t),y(t),z(t))\rangle,$$ 
$a \leq t \leq b$:


\begin{align*}
 \int_C F \, ds &= \int_a^b F(x(t),y(t),z(t)) ds \\
\intertext{where}
 ds 				&= \sqrt{x'(t)^2 + y'(t)^2 + z'(t)^2} \, dt 
\end{align*}

\end{minipage}
}
\quad
\adjustbox{valign=t}
{
\begin{minipage}{0.45\textwidth}
\footnotesize
\begin{center}
\textbf{Scalar Surface Integrals}
\end{center}

\medskip

\scriptsize

If $S$ is parameterized by
$$\bfr(u,v) = \langle x(u,v),y(u,v),z(u,v)\rangle,$$ 
$(u,v) \in D$:

$$
 \iint_S F \, dS = 
 \iint_D F 
  |\bfr_u \times \bfr_v| \, du \, dv 
$$
where
\begin{align*}
F			&=	F(x(u,v), y(u,v), z(u,v))\\
\bfr_u 	&=	\frac{\dee}{\dee u} \bfr(u,v)\\
\bfr_v	&=	\frac{\dee}{\dee v} \bfr(u,v)
\end{align*}

\end{minipage}
}
\end{frame}

\begin{frame}
\frametitle{Sneak Preview - Vector Surface Integrals}
\adjustbox{valign=t}
{
\begin{minipage}{0.45\textwidth}
\footnotesize
\begin{center}
\textbf{Vector Line Integrals}
\end{center}

\medskip

\scriptsize
If $C$ is parameterized by 
$$\bfr(t) = \langle x(t),y(t),z(t))\rangle,$$ 
$a \leq t \leq b$:

\medskip


\begin{align*}
\int_C \bfF \cdot \, d\bfr 	&=  \int_a^b \bfF \cdot \, \bfT(t) \, ds  \\
\\
\bfF 								&= \bfF(x(t),y(t),z(t)) \\
 \bfT(t) 							&= \frac{\bfr'(t)}{|\bfr'(t)|} 
\end{align*}
\end{minipage}
}
\quad
\adjustbox{valign=t}
{
\begin{minipage}{0.45\textwidth}
\footnotesize
\begin{center}
\textbf{Vector Surface Integrals}
\end{center}

\scriptsize

If $S$ is parameterized by 
$$\bfr(u,v) = \langle x(u,v),y(u,v),z(u,v)\rangle,$$
$(u,v) \in D$:

\medskip


\begin{align*}
\iint_S \bfF \cdot{d \mathbf{S}}	&= 	\iint_D \bfF \cdot \bfN \, du \, dv\\
\\
\bfF	&=	\bfF(x(u,v),y(u,v),z(u,v))\\
\bfN	&=	\bfr_u \times \bfr_v 
\end{align*}

\end{minipage}
}
\end{frame}

\section{Review}

\begin{frame}
\frametitle{Parameterized Surface Review}
%\adjustbox{valign=t}
{
	\begin{minipage}{0.35\textwidth}
	\begin{center}

	%%%
	%%%		Graphic 1 - the uv region for the sphere
	%%%
	
	\begin{tikzpicture}[scale=0.7]
	%%
	%%	Draw the grid
	%%
	\draw[help lines,step=0.5]	(0,0)	grid	(4,2);
	\draw[thick,->,>=stealth]	(-0.5,0)	--	(4.5,0) 	node[anchor=west] 	{\scriptsize{$u$}};
	\draw[thick,->,>=stealth]	(0,-0.5)	--	(0,2.5)	node[anchor=south]	{\scriptsize{$v$}};
	\node[anchor=north east] 	at 	(0,0)		{\scriptsize{$0$}};
	\node[anchor=east]				at	(0,2)		{\scriptsize{$\pi$}};
	\node[anchor=north]			at	(4,0)		{\scriptsize{$2\pi$}};
	\node [above] 					at	(1.125,1.015)	{\scriptsize{$D$}};
	%%
	%%	Red dot represents the point (\pi/4,\pi.4) in the parameter space
	%%
	\draw[red,fill=red]			(0.5,0.5)		circle(0.10cm);
	
	%%
	%%	Uncover line of constant v
	%%
	
	\uncover<2->
	{
		\draw[thick,red]				(0,0.50)	--	(4,0.50);
		\node[left]	at					(0,0.5)			{\scriptsize{$\pi/4$}};
		
	}
	
	%%
	%%	Uncover line of constant u
	%%
	
	\uncover<3->
	{
		\draw[thick,cyan]			(0.50,0)	--	(0.50,2);
		\node[below]	at				(0.5,0)			{\scriptsize{$\pi/4$}};
		
		%	remove ugly cyan line through the point
		\draw[red,fill=red]			(0.5,0.5)		circle(0.10cm);
	}
	
	\end{tikzpicture}
	
	\vspace{-0.25cm}
	
	{\tiny{The parameter space $D$}}
	
	\bigskip
	

	
	
	%%
	%%	Graphic 2 - The sphere
	%%
	
	\tdplotsetmaincoords{65}{110}
	\begin{tikzpicture}[scale=0.5,tdplot_main_coords]
	
	%%
	%%	1. 	Draw sphere
	%%
	
	\foreach \u in {7,17,...,367}
		{
			\draw[gray]
				plot[domain=0:180,variable=\v]
					(
						{3*sin(\v)*cos(\u)},
						{3*sin(\v)*sin(\u)},
						{3*cos(\v)}
					);
		}
	\foreach \v in {0,10,...,180}
		{	
			\draw[gray]
				plot[domain=0:360,variable=\u]
					(
						{3*sin(\v)*cos(\u)},
						{3*sin(\v)*sin(\u)},
						{3*cos(\v)}
					);
		}
		
		%% add red dot at $(\pi/4,\pi/4)$
		
		\draw[red,fill=red]	(
										{3*sin(45)*cos(45)},
										{3*sin(45)*sin(45)},
										{3*cos(45)}
									)
					circle(0.10cm);	
					
		%%
		%%	2. Show curve of constant v 
		%%
		
		\uncover<2->
		{
			\draw[red,dashed]
				plot[domain=0:360,variable=\theta]
					(
						{3*sin(45)*cos(\theta)},
						{3*sin(45)*sin(\theta)},
						{3*cos(45)}
					);
		}
					
		%%
		%%	3. Show curve of constant u and redraw dot
		%%		so that it doesn't have an ugly cyan line
		%%		through it
		%%
		
		\uncover<3->
		{
			\draw[cyan,dashed]
				plot[domain=0:180,variable=\phi]
					(
						{3*sin(\phi)*cos(45)},
						{3*sin(\phi)*sin(45)},
						{3*cos(\phi)}
					);
					
			\draw[red,fill=red]	(
										{3*sin(45)*cos(45)},
										{3*sin(45)*sin(45)},
										{3*cos(45)}
									)
					circle(0.10cm);	
		}
					
		%%
		%%	4. Add tangent vectors
		%%
		
		\uncover<4->
		%% tangent vectors
		{
			\draw[red,very thick,->,>=stealth]
				 	(
				 			{3*sin(45)*cos(45)},
				 			{3*sin(45)*sin(45)},
				 			{3*cos(45)}
				 	)	--
				 ++(-1,1,0);
			\draw[cyan,very thick,->,>=stealth]
				 	(
				 		{3*sin(45)*cos(45)},
				 		{3*sin(45)*sin(45)},
				 		{3*cos(45)}
				 	)	--
				 ++(1,1,-{sqrt(2)});
		}
		
		%%
		%%	5.	Add tangent plane
		%%
		
		\uncover<5->
		{
			\draw[orange,thick,fill=orange!20,opacity=0.5]
					(
				 			{3*sin(45)*cos(45)},
				 			{3*sin(45)*sin(45)-2},
				 			{3*cos(45)+sqrt(2)}
				 	)
				 	--
				 	++(-2,2,0)
				 	--
				 	++(2,2,{-2*sqrt(2)})
				 	--
				 	++(2,-2,0)
				 	--
				 	++(-2,-2,{2*sqrt(2)});
					
		}
		
		%%
		%%	6. Add outward normal
		%%
		
		\uncover<6->
		{
			\draw[very thick,black,->,>=stealth]
					(
				 		{3*sin(45)*cos(45)},
				 		{3*sin(45)*sin(45)},
				 		{3*cos(45)}
				 	)	--	++(-3,-3,-{3*sqrt(2)});
		}
	\end{tikzpicture}
	
	\smallskip
	
	\tiny{$\sin v \cos u \ii + \sin v \sin u \jj + \cos v \kk$}
	\end{center}
	\end{minipage}
}
\quad
%\adjustbox{valign=t}
{
\begin{minipage}{0.55\textwidth}
\footnotesize
\uncover<1->

A \emph{parameterized surface} is traced out by $\bfr(u,v)$ where $(u,v) \in D$, a region in the $uv$ plane. 

\medskip
\uncover<2->
{
If $v$ is held fixed and $u$ varies, the result is a curve along the surface.
}

\medskip
\uncover<3->
{
If $u$ is held fixed and $v$ varies, the result is a different curve along the surface.
}

\medskip
\uncover<4->
{
Each curve has a tangent vector, so there are two independent tangent vectors
$${\color{red} \bfr_u = \dee \bfr/\dee u}, \quad {\color{cyan} \bfr_v = \dee \bfr /\dee v}$$ 
}

\uncover<5->
{
The vectors $\bfr_u$ and $\bfr_v$  span a 
tangent plane
}

\medskip
\uncover<6->
{
The normal to the tangent plane is $$\bfN = \bfr_u \times \bfr_v$$
\end{minipage}
}

}
\end{frame}

\section{Scalar Surface Integrals}

\begin{frame}
\frametitle{Scalar Surface Integrals}
\footnotesize
If $S$ is parameterized by $\bfr(u,v)$ for $(u,v) \in D$, and $f$ is a function continuous in a neighborhood of $S$,
$$ \iint_S f(x,y,z) \, dS = \iint_D F(x(u,v),y(u,v),z(u,v)) \, \left| \bfr_u \times \bfr_v \right| \, du \, dv $$

\bigskip
\begin{enumerate}
\item	Suppose that $f(x,y,z) = g(\sqrt{x^2+y^2+z^2})$ where $g(2) = 5$. Find $\iint_S f(x,y,z) \, dS$ if $S$ is the sphere $x^2 + y^2 + z^2 = 4$.

\smallskip

\item	Find $\iint_S xz \, dS$ if $S$ is the part of the plane $2x+2y+z=4$ that lies in the first octant.

\smallskip

\item	Find $\iint y^2 \, dS$ if $S$ is the part of the sphere $x^2+y^2+z^2=1$ that lies above the cone $z=\sqrt{x^2+y^2}$.
\end{enumerate}
\end{frame}

\section{Oriented Surfaces}


\begin{frame}
\frametitle{The Oriented Unit Normal to a Surface}
\footnotesize
A surface $S$ is called an  \emph{oriented surface} if there is a unit normal vector $\bfn$ at every point on the surface that varies continuously along the surface. Every parameterized surface has such a unit normal, given by
$$ \bfn = \frac{\bfr_u \times \bfr_v}{\left| \bfr_u \times \bfr_v \right|}. $$

\medskip

Every orientable surface in $\R^3$ has \emph{two} possible orientations, one with $\bfn$ and the other with $-\bfn$. 

\begin{center}
\adjustbox{valign=t}
{
		\begin{minipage}{0.45\textwidth}
		\begin{center}
		\tiny{The orientation of the sphere with}
		\tiny{$\bfn = \frac{\bfr_u \times \bfr_v}{\left| \bfr_u \times \bfr_v\right|}$}
		
		\medskip
		
		\tdplotsetmaincoords{65}{110}
	\begin{tikzpicture}[scale=0.5,tdplot_main_coords]
	
	%%
	%%	1. 	Draw sphere
	%%
	
	\foreach \u in {7,17,...,367}
		{
			\draw[gray]
				plot[domain=0:180,variable=\v]
					(
						{3*sin(\v)*cos(\u)},
						{3*sin(\v)*sin(\u)},
						{3*cos(\v)}
					);
		}
	\foreach \v in {0,10,...,180}
		{	
			\draw[gray]
				plot[domain=0:360,variable=\u]
					(
						{3*sin(\v)*cos(\u)},
						{3*sin(\v)*sin(\u)},
						{3*cos(\v)}
					);
		}
		
		%% add red dot at $(\pi/4,\pi/4)$
		
		\draw[red,fill=red]	(
										{3*sin(45)*cos(45)},
										{3*sin(45)*sin(45)},
										{3*cos(45)}
									)
					circle(0.10cm);	
					
					
		%%
		%%	Add tangent vectors
		%%
		
		
		%% tangent vectors
		{
			\draw[red,very thick,->,>=stealth]
				 	(
				 			{3*sin(45)*cos(45)},
				 			{3*sin(45)*sin(45)},
				 			{3*cos(45)}
				 	)	--
				 ++(-1,1,0);
			\draw[cyan,very thick,->,>=stealth]
				 	(
				 		{3*sin(45)*cos(45)},
				 		{3*sin(45)*sin(45)},
				 		{3*cos(45)}
				 	)	--
				 ++(1,1,-{sqrt(2)});
		}
		
		
		%%
		%%	Add inward normal
		%%
		
		{
			\draw[very thick,black,->,>=stealth]
					(
				 		{3*sin(45)*cos(45)},
				 		{3*sin(45)*sin(45)},
				 		{3*cos(45)}
				 	)	--	++(-3,-3,-{3*sqrt(2)});
		}
		\end{tikzpicture}
		\end{center}
		\end{minipage}
}
\quad
\adjustbox{valign=t}
{
		\begin{minipage}{0.45\textwidth}
		\tiny{The orientation of the sphere with}
		\tiny{$\bfn = {\color{red} -} \frac{\bfr_u \times \bfr_v}{\left| \bfr_u \times \bfr_v\right|}$}
		
		\medskip
		
		\tdplotsetmaincoords{65}{110}
	\begin{tikzpicture}[scale=0.5,tdplot_main_coords]
	
	%%
	%%	1. 	Draw sphere
	%%
	
	\foreach \u in {7,17,...,367}
		{
			\draw[gray]
				plot[domain=0:180,variable=\v]
					(
						{3*sin(\v)*cos(\u)},
						{3*sin(\v)*sin(\u)},
						{3*cos(\v)}
					);
		}
	\foreach \v in {0,10,...,180}
		{	
			\draw[gray]
				plot[domain=0:360,variable=\u]
					(
						{3*sin(\v)*cos(\u)},
						{3*sin(\v)*sin(\u)},
						{3*cos(\v)}
					);
		}
		
		%% add red dot at $(\pi/4,\pi/4)$
		
		\draw[red,fill=red]	(
										{3*sin(45)*cos(45)},
										{3*sin(45)*sin(45)},
										{3*cos(45)}
									)
					circle(0.10cm);	
					
					
		%%
		%%	Add tangent vectors
		%%
		
		
		%% tangent vectors
		{
			\draw[red,very thick,->,>=stealth]
				 	(
				 			{3*sin(45)*cos(45)},
				 			{3*sin(45)*sin(45)},
				 			{3*cos(45)}
				 	)	--
				 ++(-1,1,0);
			\draw[cyan,very thick,->,>=stealth]
				 	(
				 		{3*sin(45)*cos(45)},
				 		{3*sin(45)*sin(45)},
				 		{3*cos(45)}
				 	)	--
				 ++(1,1,-{sqrt(2)});
		}
		
		
		%%
		%%	Add inward normal
		%%
		
		{
			\draw[very thick,black,->,>=stealth]
					(
				 		{3*sin(45)*cos(45)},
				 		{3*sin(45)*sin(45)},
				 		{3*cos(45)}
				 	)	--	++(3,3,{3*sqrt(2)});
		}
		\end{tikzpicture}
		\end{minipage}
}
\end{center}
\end{frame}

\begin{frame}
\frametitle{Oriented Surfaces versus the M\"{o}bius Strip}
\begin{minipage}{0.45\textwidth}
\footnotesize
\begin{center}
\includegraphics[width=0.8\textwidth]{Images/MobiusBand.jpg}

\smallskip

The M\"{o}bius Band
\end{center}
\end{minipage}
\quad
\begin{minipage}{0.45\textwidth}
\footnotesize
\begin{center}
\includegraphics[width=0.8\textwidth]{Images/Mobius_2.jpg}
\end{center}

\smallskip

August Ferdinand M\"{o}bius (1790--1868)\\
\end{minipage}

\end{frame}


\section{Vector Surface Integrals}

\begin{frame}
\frametitle{Vector Surface Integrals}
\footnotesize
If $\bfF$ is a continuous vector field defined on an oriented surface $S$ with unit normal vector $\bfn$, the \emph{surface integral of $\bfF$ over $S$} is
$$ \iint_S \bfF \, d \bfS = \iint_S \bfF \cdot \bfn \, dS $$

\medskip
This integral is also called the \emph{flux} of $\bfF$ across $S$. Depending on the choice of normal, it measures either what goes \emph{in} (inward normal) or what comes \emph{out} (outward normal).

\bigskip

\begin{enumerate}
% 16.7.25
\item	Find the flux of $\bfF(x,y,z) = x \ii +  y \jj + z^2 \kk$ across $S$ if $S$ is the sphere of radius $1$ and center at the origin

\smallskip

% 16.7.27
\item	Find the flux of $\bfF(x,y,z) = y \jj - z \kk$ across the paraboloid $y=x^2 + z^2$, $0 \leq y \leq 1$, and the disc $x^2 + y^2 \leq 1$, $y=1$
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