From c46724301a1359c35e1ea2f49c44b5b13daa2341 Mon Sep 17 00:00:00 2001 From: Worker Agent Date: Thu, 4 Jun 2026 19:30:31 -0500 Subject: [PATCH] polish: cross-reference verification and final formatting - Fix overfull hbox in ch05 method comparison table (118pt -> resolved) - Fix overfull hbox in ch09 verification equation (147pt -> split with align) - Fix overfull hbox in ch08 phase plane table (16pt -> p-column width) - Fix overfull hbox in appA VOP formula (70pt -> inline textstyle) - Fix overfull hbox in appB Fourier series table (70pt -> resizebox) - Fix overfull hbox in appB common integrals table (tabular width) - Fix overfull hbox in appB Laplace properties table (tabular width) - Fix float too large in ch06 TikZ resonance figure (scale + clip) - Fix text overfull in ch03 terminal velocity paragraph - Fix ch09 summary table width with @{} column specifiers --- appendices/appA_solution_summary.tex | 8 +++----- appendices/appB_transform_tables.tex | 13 ++++++++----- chapters/ch03_qualitative.tex | 2 +- chapters/ch05_second_order_nonhomogeneous.tex | 8 ++++---- chapters/ch06_mechanical_applications.tex | 8 +++++--- chapters/ch08_systems.tex | 2 +- chapters/ch09_series_solutions.tex | 14 ++++++++------ 7 files changed, 30 insertions(+), 25 deletions(-) diff --git a/appendices/appA_solution_summary.tex b/appendices/appA_solution_summary.tex index f599aa9..7666d19 100644 --- a/appendices/appA_solution_summary.tex +++ b/appendices/appA_solution_summary.tex @@ -89,11 +89,9 @@ Undetermined Coefficients & See \cref{tab:appA_undetermined_guess} for guess table. Plug $y_p$ into ODE, solve for unknown coefficients. & $g(x)$ is a polynomial, exponential, sine/cosine, or finite sums/products thereof \\[12pt] Variation of Parameters & -\[ - y_p(x) = -y_1(x)\int \frac{y_2(x)\,g(x)}{a\,W(x)}\,\diff x - + y_2(x)\int \frac{y_1(x)\,g(x)}{a\,W(x)}\,\diff x -\] -where $W = y_1 y_2' - y_2 y_1'$. & +$\textstyle y_p(x) = -y_1(x)\!\int \frac{y_2(x)\,g(x)}{a\,W(x)}\,\diff x + y_2(x)\!\int \frac{y_1(x)\,g(x)}{a\,W(x)}\,\diff x$ + +where $\textstyle W = y_1 y_2' - y_2 y_1'$. & Any $g(x)$ for which the integrals can be evaluated; requires the fundamental set $\{y_1,y_2\}$ \\[8pt] \bottomrule \end{tabular} diff --git a/appendices/appB_transform_tables.tex b/appendices/appB_transform_tables.tex index c4d14ef..352b802 100644 --- a/appendices/appB_transform_tables.tex +++ b/appendices/appB_transform_tables.tex @@ -21,7 +21,7 @@ provided the integral converges. The following table lists the most frequently e \centering \caption{Common Laplace Transforms} \label{tab:laplace_transforms} -\begin{tabular}{l l l} +\begin{tabular}{l l p{3cm}} \toprule \textbf{$f(t)$} & \textbf{$\mathcal{L}\{f(t)\} = F(s)$} & \textbf{Conditions} \\ \midrule @@ -56,7 +56,7 @@ The following algebraic and operational properties make the Laplace transform a \centering \caption{Laplace Transform Properties} \label{tab:laplace_properties} -\begin{tabular}{l l} +\begin{tabular}{@{}l p{12cm}@{}} \toprule \textbf{Property} & \textbf{Formula} \\ \midrule @@ -90,11 +90,13 @@ Fourier series decompose a periodic function $f(x)$ of period $2L$ into sine and \centering \caption{Fourier Series on $[-L, L]$} \label{tab:fourier_series} +\renewcommand{\arraystretch}{1.3} +\resizebox{\textwidth}{!}{% \begin{tabular}{l l} \toprule \textbf{Formula} & \textbf{Expression} \\ \midrule -Full series & $\displaystyle f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \Bigl[a_n \cos\!\Bigl(\frac{n\pi x}{L}\Bigr) + b_n \sin\!\Bigl(\frac{n\pi x}{L}\Bigr)\Bigr]$ \\[12pt] +Full series & $\displaystyle f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \Bigl[a_n \cos\!\Bigl(\frac{n\pi x}{L}\Bigr) + b_n \sin\!\Bigl(\frac{n\pi x}{L}\Bigr)\Bigr]$ \\[10pt] $a_0$ & $\displaystyle a_0 = \frac{1}{L}\int_{-L}^{L} f(x)\,\diff x$ \\[8pt] $a_n$ & $\displaystyle a_n = \frac{1}{L}\int_{-L}^{L} f(x)\cos\!\Bigl(\frac{n\pi x}{L}\Bigr)\,\diff x$ \\[8pt] $b_n$ & $\displaystyle b_n = \frac{1}{L}\int_{-L}^{L} f(x)\sin\!\Bigl(\frac{n\pi x}{L}\Bigr)\,\diff x$ \\[8pt] @@ -104,7 +106,8 @@ Complex coefficients & $\displaystyle c_n = \frac{1}{2L}\int_{-L}^{L} f(x)\,e^{- Complex series & $\displaystyle f(x) = \sum_{n=-\infty}^{\infty} c_n\,e^{i n\pi x/L}$ \\[8pt] Parseval's identity & $\displaystyle \frac{1}{L}\int_{-L}^{L} |f(x)|^{2}\,\diff x = \frac{a_0^{2}}{2} + \sum_{n=1}^{\infty} \bigl(a_n^{2} + b_n^{2}\bigr)$ \\[8pt] \bottomrule -\end{tabular} +\end{tabular}% +} \end{table} \subsection{Common Integral Table} @@ -116,7 +119,7 @@ The following integrals are used throughout the handbook, particularly in separa \centering \caption{Common Indefinite Integrals} \label{tab:common_integrals} -\begin{tabular}{l l} +\begin{tabular}{@{}l p{11.5cm}@{}} \toprule \textbf{Integrand} & \textbf{Result} \\ \midrule diff --git a/chapters/ch03_qualitative.tex b/chapters/ch03_qualitative.tex index 4f5123e..9449e78 100644 --- a/chapters/ch03_qualitative.tex +++ b/chapters/ch03_qualitative.tex @@ -612,7 +612,7 @@ Since $f'(v^*) = -k/m < 0$, the terminal velocity is asymptotically stable: any \label{eq:falling_quadratic} m\,\frac{\mathrm{d}v}{\mathrm{d}t} = mg - kv^2. \end{equation} -The terminal velocity is $v^* = \sqrt{mg/k}$. The qualitative behavior is the same---solutions converge to $v^*$ asymptotically---but the approach is different from the linear case. +The terminal velocity is $v^* = \sqrt{mg/k}$. The qualitative behavior is the same: solutions converge to $v^*$ asymptotically. The approach, however, is different from the linear case. \subsection{Summary} \label{sec:ch03_summary} diff --git a/chapters/ch05_second_order_nonhomogeneous.tex b/chapters/ch05_second_order_nonhomogeneous.tex index 74b6005..46f73a1 100644 --- a/chapters/ch05_second_order_nonhomogeneous.tex +++ b/chapters/ch05_second_order_nonhomogeneous.tex @@ -723,17 +723,17 @@ Both undetermined coefficients (UC) and variation of parameters (VoP) find parti \centering \caption{Undetermined coefficients vs.\ variation of parameters} \label{tab:ch05_method_comparison} -\begin{tabular}{l l l} +\begin{tabular}{p{2.8cm} p{4.8cm} p{6.2cm}} \toprule \textbf{Aspect} & \textbf{Undetermined Coefficients} & \textbf{Variation of Parameters} \\ \midrule - Applicable $g(x)$ & Polynomials, $e^{\alpha x}$, $\sin(\beta x)$, $\cos(\beta x)$, and their sums/products & \emph{Any} continuous $g(x)$ (in principle) \\ - Requires $y_h$? & Only for overlap check (modification rule) & Yes --- need a full fundamental set \\ + Applicable $g(x)$ & Polynomials, $e^{\alpha x}$, $\sin/\cos(\beta x)$, sums/products & \emph{Any} continuous $g(x)$ (in principle) \\ + Requires $y_h$? & Only for overlap check & Yes --- need a full fundamental set \\ Requires $W(x)$? & No & Yes \\ Normalized form? & Not required & \textbf{Required} (coefficient of $y''$ must be~1) \\ Computation & Algebra (solve linear system) & Integration (may be difficult or non-elementary) \\ Speed (simple $g(x)$) & Fast --- algebraic, no integrals & Slower --- integrals needed \\ - Speed (complex $g(x)$) & \textbf{Not applicable} --- method fails & Always applicable (if integrals exist) \\ + Speed (complex $g(x)$) & \textbf{N/A} --- method fails & Always applicable (if integrals exist) \\ \bottomrule \end{tabular} \end{table} diff --git a/chapters/ch06_mechanical_applications.tex b/chapters/ch06_mechanical_applications.tex index acf7e69..6a5c98a 100644 --- a/chapters/ch06_mechanical_applications.tex +++ b/chapters/ch06_mechanical_applications.tex @@ -492,7 +492,7 @@ The amplitude $C(\omega)$ as a function of the forcing frequency is called the \ \begin{figure}[htbp] \centering -\begin{tikzpicture}[scale=0.9] +\begin{tikzpicture}[scale=0.8, clip] \draw[->] (0,0) -- (6.5,0); \draw[->] (0,-0.3) -- (0,3.5); \node[below] at (6.5,0) {\small $\omega$}; @@ -507,8 +507,10 @@ The amplitude $C(\omega)$ as a function of the forcing frequency is called the \ \draw[thick, blue!70] plot[domain=0.1:6.2, samples=300, smooth] ({\x}, {2.0 / sqrt((1 - 0.0816*\x*\x)^2 + (0.15*\x)^2)}); - % Undamped curve (narrower, higher peak) - \draw[thick, red!60, dashed] plot[domain=0.1:6.2, samples=300, smooth] + % Undamped curve (narrower, higher peak) -- split around singularity at omega~3.51 + \draw[thick, red!60, dashed] plot[domain=0.1:3.35, samples=150, smooth] + ({\x}, {2.0 / abs(1 - 0.0816*\x*\x)}); + \draw[thick, red!60, dashed] plot[domain=3.65:6.2, samples=150, smooth] ({\x}, {2.0 / abs(1 - 0.0816*\x*\x)}); % Legend diff --git a/chapters/ch08_systems.tex b/chapters/ch08_systems.tex index 24b1127..ce031cd 100644 --- a/chapters/ch08_systems.tex +++ b/chapters/ch08_systems.tex @@ -518,7 +518,7 @@ For $2 \times 2$ systems $\mathbf{x}' = A\mathbf{x}$, the qualitative behavior o \textbf{Phase plane classification for $2 \times 2$ systems $\mathbf{x}' = A\mathbf{x}$.} \begin{center} - \begin{tabular}{l l} + \begin{tabular}{l p{10cm}} \toprule \textbf{Eigenvalues} & \textbf{Classification} \\ \midrule diff --git a/chapters/ch09_series_solutions.tex b/chapters/ch09_series_solutions.tex index e8cefa5..e23e64e 100644 --- a/chapters/ch09_series_solutions.tex +++ b/chapters/ch09_series_solutions.tex @@ -317,9 +317,11 @@ The repeated-root case is analogous to the repeated-root case for constant-coeff \textit{Verification.} Compute $y' = 3c_1 x^2 - c_2 x^{-2}$ and $y'' = 6c_1 x + 2c_2 x^{-3}$. Substitute into the ODE: \[ - x^2(6c_1 x + 2c_2 x^{-3}) - x(3c_1 x^2 - c_2 x^{-2}) - 3(c_1 x^3 + c_2 x^{-1}) - = 6c_1 x^3 + 2c_2 x^{-1} - 3c_1 x^3 + c_2 x^{-1} - 3c_1 x^3 - 3c_2 x^{-1} - = (6 - 3 - 3)c_1 x^3 + (2 + 1 - 3)c_2 x^{-1} = 0. \quad \checkmark + \begin{split} + & x^2(6c_1 x + 2c_2 x^{-3}) - x(3c_1 x^2 - c_2 x^{-2}) - 3(c_1 x^3 + c_2 x^{-1}) \\ + & \quad = 6c_1 x^3 + 2c_2 x^{-1} - 3c_1 x^3 + c_2 x^{-1} - 3c_1 x^3 - 3c_2 x^{-1} \\ + & \quad = (6 - 3 - 3)c_1 x^3 + (2 + 1 - 3)c_2 x^{-1} = 0. \quad \checkmark + \end{split} \] \end{workedexample} @@ -578,16 +580,16 @@ Cases~2 and~3 involve the logarithmic term $\ln(x)$, analogous to the repeated-r \centering \caption{Series solution methods: when to use and solution forms} \label{tab:ch09_summary} -\begin{tabular}{l l p{4.5cm}} +\begin{tabular}{@{}l l p{4.2cm}@{}} \toprule \textbf{Method} & \textbf{When to use} & \textbf{Solution form} \\ \midrule Power series & $p(x)$ and $q(x)$ analytic at $x_0$ & $y(x) = \sum_{n=0}^{\infty} a_n (x-x_0)^n$ \\[12pt] Euler--Cauchy & $x^2 y'' + \alpha x y' + \beta y = 0$ & -$y = c_1 x^{r_1} + c_2 x^{r_2}$ (and variants, see \cref{sec:ch09_euler_cauchy}) \\[12pt] +$y = c_1 x^{r_1} + c_2 x^{r_2}$ (see \cref{sec:ch09_euler_cauchy}) \\[12pt] Frobenius & $x_0$ is a regular singular point; $xp(x)$ and $x^2q(x)$ analytic at $x_0$ & -$y(x) = x^r \sum_{n=0}^{\infty} a_n x^n$ (with $r$ from the indicial equation) \\ +$y(x) = x^r \sum_{n=0}^{\infty} a_n x^n$ \\ \bottomrule \end{tabular} \end{table}