TASK
Rewrite text in symbolic language.
REQUIREMENTS:
Katex math requirements
- Use $...$/$$...$$ delimiters
- $$ should occupy its own line
- Avoid using $'$ for for particular value of a variable to avoid ambiguity
- use $_1$, $_2$...
- Use $\dot{}$, $\ddot{}$, ... for derivatives
GOOD:
$$
\Phi(\mathbf r_1)=-G\int\frac{\rho(\mathbf r)}{|\mathbf r_1-\mathbf r|}d^3r_1
$$
$\ddot{\mathbf r}=-G(M+m)\frac{\mathbf r}{r^3}$
BAD:
$$\Phi(\mathbf r)
=-G\int\frac{\rho(\mathbf r')}{|\mathbf r-\mathbf r'|}d^3r'$$
Use $\vec u$ for vectors and $\mathbf{M}$ for matrices, not simply $u$ or $M$ for clarity
Use ${}$
GOOD:
\mathbf{r}
BAD:
\mathbf r
- Use cases for parallel equations
GOOD:
$$
\begin{cases}
\ddot{\mathbf{r}}_m=-GM\frac{\mathbf{r}}{r^3} \text{ (1)/m} \\
\ddot{\mathbf{r}}_M=+Gm\frac{\mathbf{r}}{r^3} \text{ (2)/M}
\end{cases}
$$
BAD:
$$
\ddot{\mathbf{r}}_m=-GM\frac{\mathbf{r}}{r^3} \text{ (1)/m} \\
\ddot{\mathbf{r}}_M=+Gm\frac{\mathbf{r}}{r^3} \text{ (2)/M}
$$
- Use symbolic language whenever possible
GOOD:
$$\exists a \in B$$
BAD:
There exists $a$ in $B$
- to emphasise multiple variables all have a value $a$, use , to separate the variables
GOOD:
$$ c_1, c_2, \cdots, c_n = 0. $$
BAD:
c_1 = c_2 = \cdots = c_n = 0.
strip any $\boxed$
Respond uniquely with the requested content. No word padding or extra explanation.
Preserve formatting
- If the original text is in a particular format (e.g., wikitext), keep it in that format.
Preserve structure
- If original text is a list, preserve the list structure