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\begin{document}

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


\title{Math 213 - Space Curves}			%%	title page
\author{Peter A. Perry}
\institute{University of Kentucky}
\date{September 13, 2019}
\maketitle


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

\begin{frame}
\frametitle{Reminders}

\begin{itemize}

\item	Access your WebWork account \emph{only through Canvas}!

\medskip

\item	Homework A4 on section 12.6 is due tonight
\item	The Review Session for Exam 1 will take place on Monday, September
		16 from 6 PM to 8 PM in KAS 213
\item	On Wednesday September 18 we will have an in-class review for Exam I
\item	Exam 1 takes place next Wednesday, September 18. Section 17 will meet in 
		CB 118, and sections 18 and 19 will meet in CB 122.

\medskip
		
	
\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%	Unit Outline
%
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\begin{frame}
\frametitle{Unit I: Geometry and Motion in Space}
\begin{itemize}

\item[{12.1}]		Lecture 1: Three-Dimensional Coordinate Systems
\item[{12.2}] 		Lecture 2: Vectors in the Plane and in Space
\item[{12.3}]		Lecture 3:The Dot Product
\item[{12.4}]		Lecture 4:The Cross Product
\item[{12.5}]		Lecture 5:	Equations of Lines and Planes, I
\item[{12.5}]		Lecture 6: Equations of  Lines and Planes, II
\item[{12.6}]		Lecture 7: Surfaces in Space
\begingroup
\color{red}
\item[{13.1}]		Lecture 8: Vector Functions and Space Curves
\endgroup
\item[{13.2}]		Lecture 9 Derivatives and Integrals of Vector Functions
\item[{}]				Lecture 10:  Exam I Review
\end{itemize}
\end{frame}

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%
%	Learning Goals
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Learning Goals}
\begin{itemize}
\item		Understand what a \textbf{vector-valued function} is
\item		Understand limits and continuity for vector-valued functions
\item		Learn to visualize space curves:
		
			\medskip
			
			 (i)  by computing their projections onto the $xy$, $xz$, and $yz$ planes,
			 
			\smallskip
			 
			 (ii) by viewing them as intersections of surfaces
\end{itemize}
\end{frame}

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%		Make font size on all slides small
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\begingroup
\small

\section{Vector-Valued Functions}

\begin{frame}
\frametitle{Vector-Valued Functions}
\footnotesize
\fcolorbox{cyan}{white}
{
\begin{minipage}{0.85\textwidth}
A \textbf{vector-valued function} is a function $\bfr(t)$ whose \emph{domain} is a set of real numbers and whose \emph{range}  is a set of vectors in two- or three-dimensional space. We can specify $\bfr(t)$ through its \emph{component functions}:
$$\bfr(t) = \langle f(t), g(t), h(t) \rangle = f(t) \ii + g(t) \jj + h(t) \kk $$
\end{minipage}
}


\vfill

Example you already know: If $\bfr_0 = \langle x_0, y_0, z_0 \rangle$ and $\bfv = \langle a,b,c \rangle$, 
then 
$$ \bfr(t)  = \bfr_0 + t \bfv $$
is a vector-valued function with component functions
$$ f(t)  = x_0 + at, \quad g(t) = y_0 + bt, \quad h(t) = z_0 + ct $$

\end{frame}

\begin{frame}
\frametitle{Vector Function Basics}
\footnotesize
\begin{enumerate}
\item	What is the domain of the function	
		$ \bfr(t) = \left\langle \ln(t+1), \dfrac{t}{9-t^2}, 2^t \right\rangle$?
\item	What is 
		$\lim_{t \to 1} \left( \dfrac{t^2-t}{t-1} \ii + \sqrt{t+8}\, \jj + \dfrac{\sin \pi t}{\ln t} \kk \right)$?
\item	Can you match these curves with their graphs?
		\begin{enumerate}
		\item[(a)]	$\bfr(t)= \langle t \sin t, t \cos t, t \rangle$
		\item[(b)]	$x=\cos t, y=\sin t, z = \cos 2t$
		\item[(c)]	$x=\cos t, y=t, z=\sin t$
		\end{enumerate}
\end{enumerate}
\begin{minipage}{0.25\textwidth}
\begin{tikzpicture}[scale=0.3]
\drawxyz{5}
\draw[red] plot[domain=0:14,smooth]	( {0.25*\x}, {sin(\x r)},{cos(\x r)});
\node at (6,0,5)		{\footnotesize{I}};
\end{tikzpicture}
\end{minipage}
\qquad
\begin{minipage}{0.25\textwidth}
\begin{tikzpicture}[scale=0.3]
\drawxyz{5}
\draw[red]	 plot[domain=0:15,smooth]	({0.25*\x*cos(\x r)}, {0.25*\x}, {0.25*\x*sin(\x r)}); 
\node at (6,0,5)		{\footnotesize{II}};
\end{tikzpicture}
\end{minipage}
\qquad
\begin{minipage}{0.25\textwidth}
\begin{tikzpicture}[scale=0.3]
\drawxyz{5}
\draw[red] plot[domain=0:6.28318,smooth]	({2*sin(\x r)}, {2*cos(2*\x r)}, {2*cos(\x r)});
\node at (6,0,5)		{\footnotesize{III}};
\end{tikzpicture}
\end{minipage}
\end{frame}


\section{Limits and Continuity}

\begin{frame}
\frametitle{Breaking it Down: Limits and Continuity}
\footnotesize
\fcolorbox{cyan}{white}
{
\begin{minipage}{0.85\textwidth}
The limit of a vector-valued function is the limit of the component functions:
$$ \lim_{t \to t_0} \langle x(t), y(t), z(t) \rangle 
		= \left\langle 	\lim_{t \to t_0} x(t) , 
							\lim_{t \to t_0} y(t),
							\lim_{t \to t_0} z(t)
			\right\rangle
$$
\end{minipage}
}

\vfill

\fcolorbox{cyan}{white}
{
\begin{minipage}{0.85\textwidth}
A vector-valued function $\bfr(t) = \langle x(t), y(t), z(t) \rangle$ is continuous at $t=a$ if each of the component functions $x(t)$, $y(t)$, $z(t)$ is continuous at $t=a$
\end{minipage}
}

\vfill
In short, $\bfr(t)$ is continuous at $a$ if $\lim_{t \to a} \bfr(t) = \bfr(a)$

\end{frame}

\section{Visualizing Space Curves}

\begin{frame}
\frametitle{Space Curves}
\footnotesize
If $\bfr(t) = \langle f(t), g(t), h(t) \rangle$ is a vector function defined on an interval $I$, the set of all points 
$(x(t),y(t),z(t)$ for $t$ in the interval $I$ is called a \emph{space curve} $C$. The equations
$ x=f(t), y=g(t), z=h(t)$ are called the parametric equations for $C$.

\vfill
Match each of the space curves shown with their parametric equations.

\medskip

\begin{tabular}{ll}
$\bfr(t) = \langle \sin t, t \rangle$	&
$\bfr(t) = \langle t^2 -1, t \rangle$	\\
$\bfr(t)	=	\langle t^2, t^3, t^4 \rangle$	&
$\bfr(t)	=	\langle \cos t, -\cos t, \sin t \rangle$
\end{tabular}

\vfill
\begin{minipage}{0.2\textwidth}
\begin{tikzpicture}[scale=0.5]
\drawxyz{2}
\draw[red]	plot[domain=0:1.75,smooth]		({\x*\x},{\x*\x*\x},{\x*\x*\x*\x});
\end{tikzpicture}
\end{minipage}
\quad
\begin{minipage}{0.2\textwidth}
\begin{tikzpicture}[scale=0.5]
\drawxyz{2};
\draw[red]	plot[domain=0:6.2832]	({cos(\x r)}, {-cos(\x r)}, {sin(\x r)});
\end{tikzpicture}
\end{minipage}
\quad
\begin{minipage}{0.2\textwidth}
\begin{tikzpicture}[scale=0.12]
\drawxy{8}
\draw[red]	plot[domain=-6:6]	({sin(\x r)},{\x});
\end{tikzpicture}
\end{minipage}
\quad
\begin{minipage}{0.2\textwidth}
\begin{tikzpicture}[scale=0.2]
\drawxy{4}
\draw[red]	plot[domain=-2:2]	({\x*\x-1},{\x});
\end{tikzpicture}
\end{minipage}

\end{frame}

\begin{frame}
\frametitle{Line Segments are Space Curves}
\footnotesize
\begin{enumerate}
\item		Find the vector equation and parametric equations for the line segment from 
$P(2,0,0)$ to $Q(6,2,-2)$

\medskip

\item		Find the vector equation and parametric equations for the line segment from 
$P(a,b,c)$ to $Q(u,v,w)$

\end{enumerate}
\end{frame}

\begin{frame}
\frametitle{Visualizing It: Projections}
\begin{minipage}{0.35\textwidth}
\begin{tikzpicture}[scale=0.6]
\drawxyz{4}
\only<2,5>
{
\draw[cyan,dashed]	plot[domain=0:6.28318]  (0,{sin(\x r)},{cos(\x r)});
}
\only<3,5>
{
\draw[blue!50,dashed]	plot[domain=0:16,smooth]	({0.25*\x},{0},{cos(\x r)});
}
\only<4,5>
{
\draw[gray,dashed]	plot[domain=0:16,smooth]	({0.25*\x},{sin(\x r)},{0});
}
\uncover<5->
{
\draw[red] plot[domain=0:16,smooth]	({0.25*\x},{sin(\x r)},{cos(\x r)});
}
\end{tikzpicture}
\end{minipage}
\qquad
\begin{minipage}{0.55\textwidth}
\footnotesize
Consider the space curve with parametric equations
$$ x(t) = \cos t, \quad y(t) = t \quad z(t) = \sin t $$
\begin{enumerate}
\uncover<2->
{
\item	Find the projection of this curve onto the $xz$ plane (the side wall)
}
\uncover<3->
{
\item	Find the projection of this curve onto the $xy$ plane (the floor)
}
\uncover<4->
{
\item	Find the projection of this curve onto the $yz$ plane (the back wall)
}
\end{enumerate}

\end{minipage}
\end{frame}

\begin{frame}
\frametitle{Visualizing It: Surfaces}
\begin{minipage}{0.35\textwidth}
\begin{tikzpicture}[scale=0.6]
\drawxyz{5}
\uncover<2->
{
\draw[cyan]	plot[domain=0:6.2832]	({0},{cos(\x r)},{sin(\x r)});
\shade[top color=blue!10,bottom color=blue!90]
			plot[domain=100:-75]		({0},{sin(\x)}, {cos(\x)})	--
			++(4,0,0)	--	
			plot[domain=-75:100]		({4},{(sin(\x)},{cos(\x)})	--
			++(-4,0,0);
\draw[cyan]	plot[domain=0:6.2832]	({4},{cos(\x r)},{sin(\x r)});
\shade[top color=white,bottom color=cyan!90,opacity=0.5]
					plot[domain=0:6.2832]	({4},{cos(\x r)},{sin(\x r)});
}
\uncover<4->
{
\draw[red,thick]					plot[domain=0:100]			({0.25*3.14159*\x/180},{sin(\x)},{cos(\x)});
\draw[red!20,thick,dashed] 	plot[domain=100:280]		({0.25*3.14159*\x/180},{sin(\x)},{cos(\x)});
\draw[red,thick]					plot[domain=280:460]		({0.25*3.14159*\x/180},{sin(\x)},{cos(\x)});
\draw[red!20,thick,dashed] 	plot[domain=460:640]		({0.25*3.14159*\x/180},{sin(\x)},{cos(\x)});
\draw[red,thick]					plot[domain=640:820]		({0.25*3.14159*\x/180},{sin(\x)},{cos(\x)});
}
\end{tikzpicture}
\end{minipage}
\qquad
\begin{minipage}{0.55\textwidth}
\footnotesize
Let's take another look at the curve
$$ x(t) = \cos t, \quad y(t) = t \quad z(t) = \sin t $$

\vfill
\begin{enumerate}
\uncover<2->
{
\item		Show that this curve lies on the cylinder $x^2+z^2 = 1$
}
\uncover<3->
{
\item		Sketch the curve and the surface on the same set of axes
}
\end{enumerate}
\end{minipage}
\end{frame}

\begin{frame}
\frametitle{Visualizing It: Surfaces}
\begin{minipage}{0.35\textwidth}
\begin{tikzpicture}[scale=0.35]
\drawxyz{6}
\uncover<2->
{
\draw[cyan]	plot[domain=0:360,smooth]		({4*cos(\x)},{6},{4*sin(\x)});
\shade[left color=white,right color=cyan!50,opacity=0.5]	plot[domain=0:360]		({4*cos(\x)},{6},{4*sin(\x)});
\shade[left color=blue!10,right color=blue!90]	
			(0,0,0)	--	plot[domain=-7:148]	({4*cos(\x)},{6},{4*sin(\x)})	--	(0,0,0);
}
\uncover<4->
{
\draw[red,thick]	plot[domain=0:148]		
	({0.5*(3.14159*\x/180)*cos(\x)},{0.75*3.14159*\x/180},{0.5*(3.14159*\x/180)*sin(\x)});
\draw[red!20,thick,dashed]	plot[domain=148:370]
	({0.5*(3.14159*\x/180)*cos(\x)},{0.75*3.14159*\x/180},{0.5*(3.14159*\x/180)*sin(\x)});
\draw[red,thick]				plot[domain=370:465]
	({0.5*(3.14159*\x/180)*cos(\x)},{0.75*3.14159*\x/180},{0.5*(3.14159*\x/180)*sin(\x)});
}
\end{tikzpicture}
\end{minipage}
\qquad
\begin{minipage}{0.55\textwidth}
\footnotesize
Let's revisit the curve
$$ x(t) = t \cos t, \quad y(t) = t \sin t, \quad z(t) = t $$

\vfill

\begin{enumerate}
\item		Show that this curve lies on the right circular  cone
			$z^2 = x^2 + y^2$
\item		Sketch the curve and the surface on the same set of axes.
\end{enumerate}
\end{minipage}
\end{frame}

\begin{frame}
\frametitle{Visualize It: Space Curves and Surfaces}
\footnotesize
Show that the curve with parametric equations 
$$x=\sin t, \quad y=\cos t, \quad z=\sin^2 t$$ 
is the curve of intersection of the surfaces $z=x^2$ and $x^2 +y^2 = 1$. 

\begin{minipage}{0.4\textwidth}
\begin{tikzpicture}[scale=0.5]
\drawxyz{6}
\only<2>
{
	\draw[cyan,thick]	plot[domain=-2:2]	({-2},{\x*\x},{\x});
	%% draw back surface plus some garbage
	\shade[bottom color=blue!70,top color=blue!10]		
			(-2,2,4)	--	plot[domain=2:-2]	({-2},{\x*\x},{\x})	--
		++	(6,0,0)	--	plot[domain=-2:2]	({4},{\x*\x},{\x})	--	++(-6,0,0);\
	%% draw front of surface, filling in the garbage
	\shade[bottom color=blue,top color=blue!20]	
			(-2,0,0)	--	plot[domain=0:2]	({-2},{\x*\x},{\x})	--	++(6,0,0)	--	
			plot[domain=2:0]	({4},{\x*\x},{\x})	--	++(-6,0,0);
	%% redraw the visible part of the z-axis
	\draw[gray!50,dashed]		(0,0,0)		--	(0,3.25,0);
	\draw[thick,black,->,>=stealth]	(0,3.25,0)	--	(0,6,0);
	%% draw the curve
	\draw[red!50]	plot[domain=0:180,smooth]
			({cos(\x)},{(sin(\x))*(sin(\x))},{sin(\x)});
	\draw[red!30,dashed]	plot[domain=180:360,smooth]
			({cos(\x)},{(sin(\x))*(sin(\x))},{sin(\x)});
}
\only<3>
{
	\draw[cyan,thick]	plot[domain=0:180]		({cos(\x)},{0},{sin(\x)});
	\shade[left color=blue,right color=blue!5]	
		plot[domain=-5:170] 	({cos(\x)},{0},{sin(\x)})	--	++(0,4,0)	--
		plot[domain=170:-5]	    ({cos(\x)},{4},{sin(\x)})	--	++(0,-4,0);
	\shade[left color=blue!50,right color=blue!5,opacity=0.5]
		plot[domain=0:360]			({cos(\x)},{4},{sin(\x)});
	%% draw the curve
	\draw[red!50]	plot[domain=0:180,smooth]
			({cos(\x)},{(sin(\x))*(sin(\x))},{sin(\x)});
	\draw[red!30,dashed]	plot[domain=180:360,smooth]
			({cos(\x)},{(sin(\x))*(sin(\x))},{sin(\x)});
}
\only<4>
{
%% draw both surfaces at half-strength plus the intersecting curve
%% parabolic cylinder back surface plus garbage
%% draw back surface plus some garbage
	\shade[bottom color=blue!70,top color=blue!10, opacity=0.5]		
			(-2,2,4)	--	plot[domain=2:-2]	({-2},{\x*\x},{\x})	--
		++	(6,0,0)	--	plot[domain=-2:2]	({4},{\x*\x},{\x})	--	++(-6,0,0);
%% circular cylinder
	\shade[left color=red,right color=red!5,opacity=0.5]	
		plot[domain=-5:170] 	({cos(\x)},{0},{sin(\x)})	--	++(0,4,0)	--
		plot[domain=170:-5]	    ({cos(\x)},{4},{sin(\x)})	--	++(0,-4,0);
%% parabolic cylinder
	%% draw front of surface, filling in the garbage
	\shade[bottom color=blue,top color=blue!20,opacity=0.8]	
			(-2,0,0)	--	plot[domain=0:2]	({-2},{\x*\x},{\x})	--	++(6,0,0)	--	
			plot[domain=2:0]	({4},{\x*\x},{\x})	--	++(-6,0,0);
%% restore top of circular cylinder
\draw[red!50,opacity=0.5]	plot[domain=0:360] 	({cos(\x)},{4},{sin(\x)});
\shade[left color=red!50,right color=white,opacity=0.5]	plot[domain=0:360] 	({cos(\x)},{4},{sin(\x)});
%% draw the curve
	\draw[red] plot[domain=0:180,smooth]
			({cos(\x)},{(sin(\x))*(sin(\x))},{sin(\x)});
	\draw[red!30,dashed]	plot[domain=180:360,smooth]
			({cos(\x)},{(sin(\x))*(sin(\x))},{sin(\x)});
}
\end{tikzpicture}
\end{minipage}
\qquad
\begin{minipage}{0.5\textwidth}
\begin{itemize}
\item		The surface $z=x^2$ is a cylinder with curve $z=x^2$ parallel to the $y$-axis
\item		The surface $x^2+y^2=1$ is a cylinder of radius $1$ parallel to the the $z$-axis
\item		Their intersection is the parametric curve above
\end{itemize}
\end{minipage}
\end{frame}

\begin{frame}
\frametitle{Intersections}
\begin{minipage}{0.4\textwidth}
\begin{tikzpicture}[scale=0.5]
\drawxyz{6}
\only<1>
{
\shade[shading=ball, ball color=blue,opacity=0.5]	
	plot[domain=0:360,smooth]	({2.236*sin(\x)},{2.236*cos(\x)},{0});
\draw[red]	plot[domain=-280:-140,smooth]	({cos(\x)},{(3.1416/180)*\x},{sin(\x)});
\draw[red!50,dashed] plot[domain=-140:135,smooth]	({cos(\x)},{(3.1416/180)*\x},{sin(\x)});
\draw[red]	plot[domain=135:280,smooth]	({cos(\x)},{(3.1416/180)*\x},{sin(\x)});
}
\only<2>
{
\draw[red]	plot[domain=0:1.45] ({\x*\x},{sqrt{4-\x*\x*\x*\x - 0.25*\x*\x}},{\x});
\draw[red,dashed]	plot[domain=-1.45:0] ({\x*\x},{sqrt{4-\x*\x*\x*\x - 0.25*\x*\x}},{\x});
}
\end{tikzpicture}
\end{minipage}
\qquad
\begin{minipage}{0.5\textwidth}
\footnotesize
\begin{enumerate}
\uncover<1->
{
\item		Find the points where the helix
			$$\bfr(t) = \langle \sin(t), \cos(t), t \rangle$$ 
			intersects the sphere
			$$x^2+y^2+z^2 = 5$$
}			
			\vfill
\uncover<2->
{			
\item		Find the  curve that describes the intersection of the parabolic cylinder $y=x^2$ 
			and the top half of the ellipsoid 
			$$x^2 + 4y^2 + 4z^2 = 16$$
}
\end{enumerate}
\end{minipage}
\end{frame}

\begin{frame}
\frametitle{Intersections}
\footnotesize
Two particles travel along the space curves
$$ \bfr_1(t) = \langle t, t^2, t^3 \rangle, \quad \bfr_2(t) = \langle 1+2t, 1 + 6t, 1 + 14t \rangle $$
Do the particles collide? Do their paths intersect?

\vfill

\emph{Recall:}

\bigskip

Two particles \emph{collide} if $\bfr_1(t) = \bfr_2(t)$ for the \emph{same} $t$.

\medskip

Two particles \emph{intersect} if $\bfr_1(s) = \bfr_2(t)$ for  (possibly different) times $s$ and $t$.
\end{frame}


\section{Summary}

\begin{frame}
\frametitle{Summary}
\footnotesize
\begin{itemize}
\item		We learned what a vector-valued function 
\item		We learned how to compute limits of space curves
\item		We learned what it means for a space curve to be continuous
\item		We learned how to visualize space curves as described by vector-valued functions
\end{itemize}

\end{frame}

\begin{frame}
\frametitle{Homework}
\begin{itemize}
\item	Begin reviewing for Exam I
\item	Re-read section 13.1
\item	Read section 13.2 for Monday
\item	Begin work on Webwork A5 which is due no later than next Wednesday
\end{itemize}
\end{frame}

\endgroup

\end{document}