Use when converting physics or mathematical physics textbook PDFs into expanded research-level tutorials with complete proofs, detailed derivations, and LaTeX-formatted PDF output
Use when converting physics or mathematical physics textbook PDFs into expanded research-level tutorials with complete proofs, detailed derivations, and LaTeX-formatted PDF output
Physics Textbook to Tutorial Converter
Transform terse physics textbook sections into comprehensive research-level tutorials with complete proofs, detailed derivations, and professional LaTeX formatting.
When to Use This Skill
Invoke this skill when:
Converting physics/math textbook PDFs to expanded tutorials
User asks to "explain", "expand", or "fill in" textbook content
Look for these gap markers that indicate content needing expansion:
Gap Marker
Meaning
Action
"Left to the reader"
Missing proof
Write complete proof
"Clearly" / "Obviously"
Skipped justification
Provide explicit reasoning
"It is easy to show"
Missing derivation
Show the full derivation
"Using Eq. (X), we get"
Skipped algebra
Show all intermediate steps
"Similarly" / "Likewise"
Pattern assumed
Write out the similar case
"Straightforward calculation"
Missing computation
Do the computation
Long equation jumps
Skipped steps
Fill in intermediate equations
Step 3: Plan the Expansion
For each section, determine:
What gaps exist?
What background might readers need?
What key insights make this topic click?
What connections to later material exist?
Step 4: Write Expanded Content
For each topic, follow this structure. Use original chapter/section numbers from the source book.
\chapter{[Original Chapter Title]} % Use same chapter number as source
\section{[Original Section Title]} % Use same section number as source (e.g., 1.3)
\sourceref{B\&F Ch.~1, \S1.3, pp.~5--8}
\subsection{Motivation and Context}
% Why do we care about this?
% What problem does it solve?
% How does it connect to what we know?
\subsection{Prerequisites}
% Brief reminder of required concepts
% Reference to earlier sections
\subsection{[Main Development]}
% Definitions first
\begin{definition}[Name]
\sourceref{Ch.~1, \S1.3, p.~5}
...
\end{definition}
% Build intuition
\begin{intuition}
...
\end{intuition}
% Formal statement
\begin{theorem}[Name]
\sourceref{Ch.~1, \S1.3, p.~6}
...
\end{theorem}
% Complete proof (cite where "left to reader" appeared)
\begin{proof}
\sourceref{Proof requested Ch.~1, \S1.3, p.~6}
...
\end{proof}
\subsection{Key Results Summary}
% Bullet points of main takeaways
Step 5: Apply Expansion Patterns
Pattern 1: "Left to the reader" proofs
Original:
"We leave the proof to the reader."
Expansion template:
\begin{proof}
We prove this in [N] steps.
\textbf{Step 1: Setup.}
[State what we're proving and introduce notation]
\textbf{Step 2: Key insight.}
[The crucial observation or technique]
\textbf{Step 3: Main derivation.}
[Detailed calculation with justification for each step]
\textbf{Step 4: Conclusion.}
[State the final result explicitly]
\end{proof}
Pattern 2: "Clearly" / "It is easy to show"
Original:
"It is easy to show that $\nabla \cdot (\nabla \times \mathbf{V}) = 0$."
Expansion:
\begin{proposition}
For any sufficiently smooth vector field $\mathbf{V}$,
\[
\nabla \cdot (\nabla \times \mathbf{V}) = 0.
\]
\end{proposition}
\begin{proof}
We compute this explicitly in Cartesian coordinates. Let $\mathbf{V} = V_x \uvect{x} + V_y \uvect{y} + V_z \uvect{z}$.
The curl is:
\[
\nabla \times \mathbf{V} =
\begin{vmatrix}
\uvect{x} & \uvect{y} & \uvect{z} \\
\partial_x & \partial_y & \partial_z \\
V_x & V_y & V_z
\end{vmatrix}
= \left(\pdv{V_z}{y} - \pdv{V_y}{z}\right)\uvect{x} + \cdots
\]
Taking the divergence:
\[
\nabla \cdot (\nabla \times \mathbf{V}) =
\pdv{}{x}\left(\pdv{V_z}{y} - \pdv{V_y}{z}\right) +
\pdv{}{y}\left(\pdv{V_x}{z} - \pdv{V_z}{x}\right) +
\pdv{}{z}\left(\pdv{V_y}{x} - \pdv{V_x}{y}\right)
\]
Expanding and grouping:
\[
= \pdv{V_z}{x}{y} - \pdv{V_y}{x}{z} +
\pdv{V_x}{y}{z} - \pdv{V_z}{y}{x} +
\pdv{V_y}{z}{x} - \pdv{V_x}{z}{y}
\]
By the equality of mixed partials (Clairaut's theorem), each term cancels with another:
\[
= \left(\pdv{V_z}{x}{y} - \pdv{V_z}{y}{x}\right) +
\left(\pdv{V_x}{y}{z} - \pdv{V_x}{z}{y}\right) +
\left(\pdv{V_y}{z}{x} - \pdv{V_y}{x}{z}\right) = 0.
\]
\end{proof}
