Some LaTex

Here are some notes I find useful for using LaTex

1. Force limits to the top and bottom side of inline sum : $\sum _{i=1}^{N}i$, use \sum\limits instead of \sum alone.
2. Force limits to the top and bottom side of inline integral : $\underset{x=1}{\overset{3}{\int }}xdx$, use \int\limits instead of \int alone.
3. Hat:
   1. Fitted value: $\hat{y}$, use \hat{}
   2. Widehat: $\widehat{\mathrm{\nabla }f}$, use \widehat{}
4. Italic: $\mathcal{L}$, it only works for upper-case letters, use \mathcal{}
5. Underset: $\underset{x}{argmin}$, use \underset{x}{argmin}
6. Gradient: $\mathrm{\nabla }f$, use \nabla
7. Field representation: the complex field $\mathbb{C}$, the real field $\mathbb{R}$, use \mathbb{}
8. Logical operator
   • P and Q: $P\wedge Q$, \land
   • P or Q: $P\vee Q$, \lor
   • not P: $\mathrm{\neg }P$, \neg, or $\overline{P}$, \bar
   • P xor Q: $P\oplus Q$, \oplus
9. Set notation
   • $\mathbb{R}$: \mathbb{R}
   • $a\in A$: a \in A
   • $a\notin A$: a \notin A
   • $\mathrm{\varnothing }$: \emptyset
   • $A\cap B$: \cap
   • $A\cup B$: \cup
   • $A\times B$: \times
   • $A\setminus B$: \setminus
   • $A\subseteq B$: \subseteq
   • $A\subset B$: \subset
   • $\bigcup _{i=1}^n$ \bigcup\limits_{i = 1}^{n}
   • $\bigcap _{i=1}^n$ \bigcap\limits_{i = 1}^{n}
10. Make equation smaller: \scriptstyle
11. Isomorphism and approximate equality
    • $G_1\cong G_2$: G_1 \cong G_2
    • $a\approx b$: a \approx b
    • $a\approxeq b$: a \approxeq b