final XeLaTeX version

Author: teepeemm

apexNotes.txt

@@ -12,7 +12,6 @@ III: F15.4.8 ?
 9.8 p529: all series tests require positivity?
 D11.4.1: Maybe give the box method for computing?
 12.5 better arc length parameterization? Frenet frame?
-F14.2.8: use axis equal image
 C15: should 2d curl match 3d curl?
 15.4: Green's Theorem on region with holes
 

errata/Errata.tex

@@ -79,20 +79,27 @@
  }
 }
 
+\setlength{\parindent}{0em}
+
 \begin{document}
 
 %\vspace*{-.6in}
 
 \maketitle
 
-\noindent
 The following errors exist in the in June 2023 printed version of Apex LT Calculus I:
 \begin{enumerate}
 \item \S1.2 p18 Example 1.2.1: The solution requires $\epsilon<4$ in order for $\delta>0$.
 \item \S3.3 p168\#25 solution: While the function is decreasing on $(-\infty,-2)$, $(-2,4)$, $(4,\infty)$; and this is the entirety of the domain; it is incorrect to say that the function is ``decreasing on [its] entire domain'' because it is not necessarily decreasing from one interval to another.
 \label{2023-06-00I}
 \end{enumerate}
 
+The following error exists in the in June 2023 printed version of Apex LT Calculus III:
+\begin{enumerate}
+\item \S15.5 p995 Example 15.5.7: The solution should switch $\phi$ and $\theta$ so that they match the notation for spherical coordinates.
+\label{2023-06-00III}
+\end{enumerate}
+
 \startMarkdownTable
 \begin{table}[ht]
 \caption{Errata Tally (``+'' indicates systemic errata)}
@@ -113,15 +120,13 @@
 \end{tabular}
 \end{table}
 
-\noindent
 The following errors exist in the in June 2021 printed version of Apex LT Calculus I:
 \begin{enumerate}
 \item \S1.5 p57 Example 7\#3a: The limit should be $-\infty$.
 \item At the back of the book, integration rule \#2 should enclose its integrand in parentheses, \#23 should have $a>0$ or use $|a|$, and \#31 should have its result in absolute values.
 \label{2021-06-00I}
 \end{enumerate}
 
-\noindent
 The following errors exist in the in June 2021 printed version of Apex LT Calculus II:
 \begin{enumerate}
 \item \S7.1 p348 \#5: The solution's line segment should go through $(5,-2)$.
@@ -159,7 +164,6 @@
 
 \clearpage
 
-\noindent
 The following errors exist in the in June 2021 printed version of Apex LT Calculus III:
 \begin{enumerate}
 \item \S12.1 p717 Example 3: In the last displayed equation, ``Multiplying $\vec r(t)$ by 5'' should produce $5\vec r(t)=\langle5\cos t+t,5\sin t+\frac32t\rangle$.
@@ -175,7 +179,6 @@
 \label{2021-06-00III}
 \end{enumerate}
 
-\noindent
 The following errors exist in the in June 2019 printed version of Apex LT Calculus I:
 \begin{enumerate}
 \item \S1.4 p41: Theorem 1.4.1 needs to say ``except possibly at $c$''.
@@ -187,7 +190,6 @@
 \label{2019-06-00Iplus}
 \end{enumerate}
 
-\noindent
 The following errors exist in the in June 2019 printed version of Apex LT Calculus II:
 \begin{enumerate}
 \item \S7.4 p364 Line -1: as $x\to\infty$, both $\sinh x$ and $\cosh x$ approach $e^x/2$.
@@ -200,7 +202,6 @@
 \label{2019-06-00II}
 \end{enumerate}
 
-\noindent
 The following errors exist in the in June 2019 printed version of Apex LT Calculus III:
 \begin{enumerate}
 \item \S11.2 p659 Definition 11.2.3 refers to $c\vec v$ as a scalar product, whereas most authors use ``scalar product'' as a synonym for the dot product.
@@ -230,7 +231,6 @@
 
 %\newpage
 
-\noindent
 The following errors exist in the July 13, 2018 printed version of Apex LT Calculus in the Important Formulas at the end of the book:
 \begin{enumerate}
 \item In Algebra / Binomial Theorem, the summation should have an index of $k$.
@@ -385,7 +385,6 @@ \section*{Digital Math Resources}
 
 In the July 27, 2017 and November 13, 2017 printed versions of Apex LT Calculus, there are numerous instances of ``$\lim A+B$''.  The convention seems to be that this should be ``$\lim(A+B)$''.  We believe that all such instances have been corrected for subsequent versions of the text.\bigskip
 
-\noindent
 The following errors exist in the November 13, 2017 printed version of Apex LT Calculus III:
 \begin{enumerate}
 \item \S11.1 p637 Example 5 Line 1: ``cylinder following cylinders'' should be ``following cylinders''.

errata/README.md

@@ -5,7 +5,7 @@ If you are interested in a particular error, you can look through [Errata.tex](E
 
 Version | Calculus I | Calculus II | Calculuc III
 ---|---|---|---
-2023-06-00|2||
+2023-06-00|2||1
 2021-06-00|2|30|10
 2019-06-00|6+|7|22
 2018-07-13|47|56+|13+
@@ -15,7 +15,7 @@ Version | Calculus I | Calculus II | Calculuc III
 2017-01-00|9+|19+|
 2016-08-00|19+||
 ---|---|---|---
-Total|96+|146+|103+
+Total|96+|146+|104+
 
 "+" indicates a systemic error.  
 (The totals for a row may be higher than what's listed in [changes.md](../changes.md) due to double counting.)

standalone.tex

@@ -25,13 +25,13 @@
 
 %\clearpage
 
-\setcounter{chapter}{1}
+\setcounter{chapter}{14}
 
 \setcounter{section}{1}
 
 %\setcounter{figure}{6}
 
-\input{text/01_Limit_Definition}
+\input{text/13_Double_Integrals_Volume}
 
 %%\printexercises{exercises/14-03-exercises}
 %

text/13_Double_Integrals_Volume.tex

@@ -384,7 +384,7 @@ \section{Double Integration and Volume}\label{sec:double_int_volume}
 Once again we make a sketch of the region over which we are integrating to facilitate changing the order. The bounds on $x$ are from $x=y$ to $x=3$; the bounds on $y$ are from $y=0$ to $y=3$. These curves are sketched in \autoref{fig:double6}, enclosing the region $R$.
 
 \mtable{Determining the region $R$ determined by the bounds of integration in \autoref{ex_double6}.}{fig:double6}{\begin{tikzpicture}
-\begin{axis}[width=1.16\marginparwidth,tick label style={font=\scriptsize},
+\begin{axis}[width=1.16\marginparwidth,tick label style={font=\scriptsize},axis equal image,
 axis y line=middle,axis x line=middle,name=myplot,axis on top,
 ymin=-.5,ymax=3.5,xmin=-.5,xmax=3.5]
 \draw [very thick,draw={\colortwo}] (axis cs:-.5,0) -- (axis cs:3.5,0)

text/14_Parametrized_Surfaces.tex

@@ -267,23 +267,23 @@ \section{Parameterized Surfaces and Surface Area}\label{sec:parametric_surfaces}
 \end{minipage}
 %
 \solution
-Recall \autoref{idea:unit_vectors} from \autoref{sec:vector_intro}, which states that all unit vectors in space have the form $\bracket{\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta}$ for some angles $\theta$ and $\varphi$. If we choose our angles appropriately, this allows us to draw the unit sphere. To get an ellipsoid, we need only scale each component of the sphere appropriately.
+Recall \autoref{idea:unit_vectors} from \autoref{sec:vector_intro}, which states that all unit vectors in space have the form $\bracket{\cos\theta\sin\varphi,\sin\theta\sin\varphi,\cos\varphi}$ for some angles $\theta$ and $\varphi$ (this also matches spherical coordinates). If we choose our angles appropriately, this allows us to draw the unit sphere. To get an ellipsoid, we need only scale each component of the sphere appropriately.
 
 The $x$-radius of the given ellipsoid is 5, the $y$-radius is 1 and the $z$-radius is 2. Substituting $u$ for $\theta$ and $v$ for $\varphi$, we have
-\[\vec r(u,v) =\bracket{5\sin u\cos v, \sin u\sin v,2\cos u},\]
+\[\vec r(u,v) =\bracket{5\cos u\sin v, \sin u\sin v,2\cos v},\]
 where we still need to determine the ranges of $u$ and $v$. 
 
-Note how the $x$ and $y$ components of $\vec r$ have  $\cos v$ and $\sin v$ terms, respectively. This hints at the fact that ellipses are drawn parallel to the $x$-$y$ plane as $v$ varies, which implies we should have $v$ range from $0$ to $2\pi$. 
+Note how the $x$ and $y$ components of $\vec r$ have  $\cos u$ and $\sin u$ terms, respectively. This hints at the fact that ellipses are drawn parallel to the $x$-$y$ plane as $u$ varies, which implies we should have $u$ range from $0$ to $2\pi$. 
 
-One may be tempted to let $0\leq u\leq 2\pi$ as well, but note how the $z$ component is $2\cos u$. We only need $\cos u$ to take on values between $-1$ and $1$ once, therefore we can restrict $u$ to $0\leq u\leq \pi$. 
+One may be tempted to let $0\leq v\leq 2\pi$ as well, but note how the $z$ component is $2\cos v$. We only need $\cos v$ to take on values between $-1$ and $1$ once, therefore we can restrict $v$ to $0\leq v\leq \pi$. 
 
 The final parameterization is thus
 \[
-\vec r(u,v) =\bracket{5\sin u\cos v, \sin u\sin v,2\cos u},
-\quad 0\leq u\leq\pi,\quad 0\leq v\leq 2\pi.
+\vec r(u,v) =\bracket{5\cos u\sin v, \sin u\sin v,2\cos v},
+\quad 0\leq u\leq2\pi,\quad 0\leq v\leq\pi.
 \]
 
-In \autoref{fig:parsurf7}(b), the ellipsoid is graphed on $\frac{\pi}{4}\leq u\leq \frac{2\pi}{3}$, $\frac{\pi}4\leq v\leq \frac{3\pi}2$ to demonstrate how each variable affects the surface.
+In \autoref{fig:parsurf7}(b), the ellipsoid is graphed on $\frac{\pi}{4}\leq u\leq \frac{3\pi}{2}$, $\frac{\pi}4\leq v\leq \frac{2\pi}3$ to demonstrate how each variable affects the surface.
 \end{example}
 
 Parameterization is a powerful way to represent surfaces. One of the advantages of the methods of parameterization described in this section is that the domain of $\vec r(u,v)$ is always a rectangle; that is, the bounds on $u$ and $v$ are constants. This will make some of our future computations easier to evaluate.