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Conventions for the solution manual

For future reference and to maintain the coherency of the solution manual, I hereby list the set of arbitrary decisions made in creating the manual.

Contents

• The only steps allowed to be skipped are purely algebraic steps; for any step involving derivatives or integrals, either it should be carried out or the derivative/integral formula being used must be explicitly stated.
• All steps within a problem are denoted using the $\Rightarrow$ symbol (\Rightarrow), except for the final answer which is to be denoted with the $\therefore$ symbol (\therefore).
• Always use $\ln$ in lieu of $\log$ to denote the natural logarithm. For any other bases, explicitly write them. eg) $\log_b x$
• Always have the differential come directly after the integral sign. eg) $\int dx f(x)$ or $\int \frac{dx}{1 + x^2}$
  • Write $\int dx f(x)$ in place of $\int^x dt f(t)$.
  • Choose a different integrating variable if it appears in the integration range.
  • Place the differential on the fraction only if it is the only term there. eg) $\int \frac{dx}{1+x^2}$, $\int dx \frac{3}{1+x^2}$
• When solving differential equations, all intermediate integration constants are denoted with any other capital alphabet than $C$ in order of appearance; those appearing in the final answer are denoted with $C$, numbered with subscripts as $C_1, C_2, \dots$ if there are multiple constants.
• Use the following Fourier transform conventions:
  • Use $k$ as the conjugate variable for $x$, and $\omega$ as that for $t$.
  • The $2\pi$ factor is equally distributed: $\hat{f}(k) = \int_{-\infty}^\infty \frac{dx}{\sqrt{2\pi}} f(x) e^{-ikx}$, $f(x) = \int_{-\infty}^\infty \frac{dk}{\sqrt{2\pi}} \hat{f}(k) e^{ikx}$
• First-order quantifiers ($\forall$, $\exists$) should appear inside parentheses, along with their binding variables and any conditions imposed on the variables. eg) $(\forall n \in \mathbb{N})$ $a_n \geq 0$
• Use of \cdot ($\cdot$) to denote multiplication of real numbers must be in order to show a clear separation of the terms.
• To define variables, either textually express the intention (eg) using the keyword "let") XOR with the symbol $:=$.
• If the left-hand side is the only newly-defined variable, then use the $:=$ symbol; otherwise, use textual indications.
  • eg) Let $v := x^3$. Suppose $\mathcal{L} u_n = \lambda_n u_n$.
  • Even in the former case, textual indications such as "let" or "suppose" are preferred if possible.
• Use the following coordinate system conventions:
  • Polar coordinates: $(r, \theta)$
  • Cylindrical coordinates: $(\rho, \theta, z)$
  • Spherical coordinates: $(r, \phi, \theta)$
    • The reversed roles of $\theta$ in the cylindrical and spherical coordinates are confusing, but this is how they are used in the textbook, so following it seems to be the best approach.

$\LaTeX$ Formatting

• When solving problems with subproblems, use the enumerate environment with the options [wide, labelindent = 0pt, label = (\alph*)].
  • Do not artificially insert a line break between the problem number and the subproblem label, nor between the subproblem label and the actual solution.
• Draw contour integrals using TikZ. Create new diagrams only when new ones are needed; that is, reuse them as much as possible.
  • The captions of the diagrams must enumerate all problems where it is used, but its label should only show the first problem.
• Line breaks must be performed with semantic considerations. Every sentence must end with a line break, and mid-sentence breaks must occur where the two parts have a definite semantic separation between them.