Tighten up language to fix changed spacing coming from new

version of latex.

CHANGES

@@ -4,7 +4,15 @@ differential equation is undefined.  While one can solve the problem
 continuously, and the formal process just works, it's best to avoid it.
 The initial condition is changed to $x(0)=\frac{\pi}{4}$ where the equation
 is well behaved.
-<li>Fix erratum from Martin Irungu, a sign error in the "proof" of Theorem
-6.1.2 (the final answer is off by a sign).  In the process, implement Martin's
-suggestion to write the whole thing in therms of the positive $s-c$ rather than
-the negative $c-s$, which clearly leads to more typos.
+<li>In the processof fixing an erratum from Martin Irungu (a sign error in the
+"proof" of Theorem 6.1.2; the final answer is off by a sign),  implement
+Martin's suggestion to write the whole thing in therms of the positive $s-c$
+rather than the negative $c-s$, which clearly leads to more typos.
+<li>For some reason a new version of LaTeX is now making some spacing slightly
+different (being a little bit more generous with space under figures for
+example), so the pagination is slightly different in places.  I went through
+and tried to get rid of bad page breaks by tightening up the language where
+needed (a good thing to do anyhow), so it is only in a couple of places where
+page breaks changed.
+<li>Some minor wording improvements.
+<li>Fix <a href="errata.html">errata</a>.

ap-linear-algebra.tex

@@ -142,8 +142,8 @@ \subsection{Vectors and operations on vectors}
 x_{1} \\ x_2 \\ \vdots \\ x_n
 \end{bmatrix} .
 \end{equation*}
-Don't worry.  It is just a different way of writing the same thing, and
-it will be useful later.  For example, the vector $(1,2)$ can be written as
+Don't worry.  It is just a different way of writing the same thing.
+For example, the vector $(1,2)$ can be written as
 \begin{equation*}
 \begin{bmatrix}
 1 \\ 2

ch-fourier-and-pde.tex

@@ -5176,11 +5176,11 @@ \section{Steady state temperature and the Laplacian}
 Y_n(y) = A_n \cosh \left( \frac{n \pi}{w} y \right)
 + B_n \sinh \left( \frac{n \pi}{w} y \right) .
 \end{equation}
-We only have one condition on $Y_n$ and hence we can pick one of $A_n$
+There is only one condition on $Y_n$ and hence we can pick one of $A_n$
 or $B_n$
 to be something convenient.
-It will be useful to have $Y_n(0) = 1$, so we let $A_n=1$.
-Setting $Y_n(h) = 0$ and solving for $B_n$ we get that
+It will be useful to have $Y_n(0) = 1$, so let $A_n=1$.
+Setting $Y_n(h) = 0$ and solving for $B_n$, we get
 \begin{equation*}
 B_n = \frac{- \cosh \left( \frac{n \pi h }{w} \right)}%
 {\sinh \left( \frac{n \pi h }{w} \right)} .
@@ -5198,9 +5198,7 @@ \section{Steady state temperature and the Laplacian}
 We define $u_n(x,y) = X_n(x)Y_n(y)$.
 And note that $u_n$
 satisfies \eqref{dirich:eq1}--\eqref{dirich:eq4}.
-
-
-Observe that
+Observe
 \begin{equation*}
 u_n(x,0) = X_n(x)Y_n(0) = \sin \left( \frac{n \pi}{w} x \right) .
 \end{equation*}
@@ -5230,12 +5228,13 @@ \section{Steady state temperature and the Laplacian}
 .
 ~~
 }
+\avoidbreak
 \end{equation*}
 As $u_n$ satisfies \eqref{dirich:eq1}--\eqref{dirich:eq4} and any linear
 combination (finite or infinite) of $u_n$ also satisfies 
 \eqref{dirich:eq1}--\eqref{dirich:eq4}, then $u$ satisfies
 \eqref{dirich:eq1}--\eqref{dirich:eq4}.
-By plugging in $y=0$, we see $u$
+We plug in $y=0$ to see $u$
 satisfies 
 \eqref{dirich:eq5} as well.
 
@@ -5249,7 +5248,7 @@ \section{Steady state temperature and the Laplacian}
 \frac{4}{n}
 \sin (n x) .
 \end{equation*}
-Therefore the solution $u(x,y)$, see \figurevref{dirichsquareplot:fig},
+The solution $u(x,y)$, see \figurevref{dirichsquareplot:fig},
 to the corresponding Dirichlet problem is
 given as
 \begin{equation*}
@@ -5483,7 +5482,7 @@ \subsection{Laplace in polar coordinates}
 is a circle rather than a rectangle.  On the other hand, what makes the
 problem somewhat more difficult is that we need polar coordinates.
 
-\begin{mywrapfigsimp}{1.3in}{1.5in}
+\begin{mywrapfigsimp}[5]{1.3in}{1.5in}
 \diffypdfversion{\vspace*{5pt}}
 \noindent
 \inputpdft{polarcoords}

ch-laplace.tex

@@ -1793,7 +1793,7 @@ \subsection{Three-point beam bending}
 \caption{Three-point bending.\label{lt:beambendingfig}}
 \end{myfig}
 
-In this case the equation becomes
+The equation becomes
 \begin{equation*}
 EI \frac{d^4 y}{dx^4} = -F \delta(x-a) ,
 \end{equation*}

ch-nonlin-systems.tex

@@ -2046,10 +2046,10 @@ \subsection{Duffing equation and strange attractors}
 
 The equation is not autonomous, so we cannot 
 draw the vector field in the phase plane.
-We can still draw the
+We can still draw
 trajectories.   
-In \figurevref{nlin:duf-two-traj} we plot trajectories for $t$ going from 0
-to 15, for two very close initial conditions
+In \figurevref{nlin:duf-two-traj}, we plot trajectories for $t$ going from 0
+to 15 for two very close initial conditions
 $(2,3)$ and $(2,2.9)$, and also the solutions in the $(x,t)$ space.  The two
 trajectories are close at first, but after a while diverge significantly.
 This sensitivity to initial conditions is precisely what
@@ -2190,8 +2190,8 @@ \subsection{Duffing equation and strange attractors}
 trajectories in \figurevref{nlin:duf-two-traj}.
 
 Let us
-compare this section to the discussion in \sectionref{forcedo:section} about forced
-oscillations.  Take the equation
+compare this section to the discussion in \sectionref{forcedo:section} on forced
+oscillations.  Consider
 \begin{equation*}
 x''+2p x' + \omega_0^2 x = \frac{F_0}{m} \cos (\omega t) .
 \end{equation*}
@@ -2218,7 +2218,7 @@ \subsection{The Lorenz system}
 is periodic or tends towards a periodic solution.  Hardly the chaotic
 behavior we are looking for.
 
-In three dimensions even autonomous systems can be chaotic.
+In three dimensions, even autonomous systems can be chaotic.
 Let us very briefly return to the \myindex{Lorenz system}
 \begin{equation*}
 x' = -10x +10y, \qquad y' = 28x-y-xz, \qquad z'=-\frac{8}{3}z + xy .

ch-power-ser.tex

@@ -365,7 +365,7 @@ \subsection{Manipulating power series}
 \sum_{k+1=1}^\infty \frac{1}{\bigl((k+1)-1\bigr)!} x^{(k+1)-1} =
 \sum_{k=0}^\infty \frac{1}{k!} x^k .
 \end{equation*}
-That was precisely the power series for $e^x$ that we started with,
+That was precisely the power series for $e^x$ we started with,
 so we showed that $\frac{d}{dx} [ e^x ] = e^x$.
 \end{example}
 
@@ -423,7 +423,7 @@ \subsection{Power series for rational functions}
 An important fact is 
 that a series for a function only defines the function
 on an interval even if the function is defined elsewhere.  For example, for
-$-1 < x < 1$ we have
+$-1 < x < 1$,
 \begin{equation*}
 \frac{1}{1-x} =
 \sum_{k=0}^\infty x^k =
@@ -1599,9 +1599,9 @@ \subsection{Bessel functions} \label{bessel:subsection}
 \begin{equation*}
 r(r-1)+r-p^2 = (r-p)(r+p) = 0 .
 \end{equation*}
-Therefore we obtain two roots $r_1 = p$ and $r_2 = -p$.
+We obtain two roots, $r_1 = p$ and $r_2 = -p$.
 If $p$ is not an integer, then following the method of Frobenius and
-setting $a_0 = 1$, we obtain
+setting $a_0 = 1$, we find
 linearly independent solutions of the form
 \begin{align*}
 & y_1 = x^p \sum_{k=0}^\infty

ch-systems.tex

@@ -2769,7 +2769,7 @@ \section{Two-dimensional systems and their vector fields}
 Let us take a moment to talk about constant coefficient linear
 homogeneous systems in the plane.
 Much intuition can be obtained by studying this simple case.
-Suppose we use coordinates $(x,y)$ for the plane as usual,
+We use coordinates $(x,y)$ for the plane as usual,
 and suppose
 $P = \left[ \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right]$ 
 is a $2 \times 2$ matrix.  Consider the system
@@ -2807,13 +2807,13 @@ \section{Two-dimensional systems and their vector fields}
 $\left[ \begin{smallmatrix} 1 \\ 1 \end{smallmatrix} \right]$.  See
 \figurevref{pln:source-eigfig}.
 
-\begin{mywrapfig}{3.25in}
+\begin{mywrapfig}[14]{3.25in}
 \capstart
 \diffyincludegraphics{width=3in}{width=4.5in}{pln-source-eig}
 \caption{Eigenvectors of $P$.\label{pln:source-eigfig}}
 \end{mywrapfig}
 
-Suppose the point $(x,y)$ is on the line determined by an eigenvector
+Let $(x,y)$ be a point on the line determined by an eigenvector
 $\vec{v}$ for an eigenvalue $\lambda$.
 That is,
 $\left[ \begin{smallmatrix} x \\ y \end{smallmatrix} \right] = \alpha \vec{v}$
@@ -2830,8 +2830,8 @@ \section{Two-dimensional systems and their vector fields}
 The derivative is a multiple of $\vec{v}$ and hence points along the
 line determined by $\vec{v}$.  As $\lambda > 0$, the derivative points in the
 direction of $\vec{v}$ when $\alpha$ is positive and in the opposite direction
-when $\alpha$ is negative.  Let us draw the lines determined by
-the eigenvectors, and let us draw
+when $\alpha$ is negative.  We draw the lines determined by
+the eigenvectors, and we draw
 arrows on the lines to indicate the directions.
 See \figurevref{pln:source-eig-arrfig}.
 
@@ -2911,11 +2911,11 @@ \section{Two-dimensional systems and their vector fields}
 \medskip
 
 \pagebreak[0]
-\emph{Case 4.} Suppose the eigenvalues are purely imaginary.
-That is, suppose the eigenvalues are $\pm ib$.  For example,
+\emph{Case 4.} Suppose the eigenvalues are purely imaginary, that is,
+$\pm ib$.  For example,
 let $P = 
 \left[ \begin{smallmatrix} 0 & 1 \\ -4 & 0 \end{smallmatrix} \right]$.
-The eigenvalues turn out to be $\pm 2i$ and eigenvectors are
+The eigenvalues are $\pm 2i$ and corresponding eigenvectors are
 $\left[ \begin{smallmatrix} 1 \\ 2i \end{smallmatrix} \right]$ and
 $\left[ \begin{smallmatrix} 1 \\ -2i \end{smallmatrix} \right]$.  Consider
 the eigenvalue $2i$ and its eigenvector
@@ -3061,12 +3061,13 @@ \subsection{Exercises}
 \task Convert this to a system of first
 order equations.
 \task Classify for what $m, c, k$ do you get which behavior.
-\task Can you explain from physical intuition why you do not get all the
+\task Explain from physical intuition why you do not get all the
 different kinds of behavior here?
 \end{tasks}
 \end{exercise}
 
 \begin{exercise}
+\pagebreak[2]
 What happens in the case when $P = 
 \left[ \begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix} \right]$?  In
 this case the eigenvalue is repeated and there is only one independent eigenvector.