A norm is actually something much more general than simply the expression you gave. It is simply a function that satisfies certain properties. However, it turns out that the $2$-norm is exactly the norm you are used to.
In general, the $p$-norm of a vector is given by
$$\|\boldsymbol {x}\|_p = \big(|x_1|^p + |x_2|^p + ... + |x_n|^p\big)^{1/p}.$$
You will see that by plugging in $p = 2$ you get the norm you are used to.
The superscript simply refers to ordinary squaring, hence
$$\|\boldsymbol{x}\|_2^2 = |x_1|^2 + |x_2|^2 + ... + |x_n|^2.$$
For you background, there is a discrepancy between notations in the U.S and some countries in Europe (mainly France and Italy, but not only). This is of course very general, there may be exceptions, but it explains why at some point the need for an unambiguous notation appeared.
Rem: I use U.S and E.U below, but again it may vary from people to people, don't be offended if you do differently...
- $\mathbb N$ in U.S does not contains $0$ while contains $0$ in E.U
Thus the unambiguous notation is $\mathbb N_0$ for $\{0,1,2,\cdots\}$.
In E.U we use $\mathbb N^*$ for $\{1,2,\cdots\}$
- Positive or negative is taken in strict sense in U.S and loose sense in E.U (this means $0$ is both a negative and a positive number).
Thus $\mathbb Z^+$ or $\mathbb R^+$ in E.U contains $0$ and not in U.S, as for the case of $\mathbb N$ we can use $\mathbb Z_0^+$ and $\mathbb R_0^+$ for expliciting the set contains $0$.
In E.U we would use $\mathbb Z^{+*}$ or $\mathbb R^{+*}$ to exclude $0$.
As you can see U.S add zero via the notation $\mathbb X_0$ while E.U exclude it via $\mathbb X^*$.
- Even with the interval notation there are differences (but only in the symbols used), U.S would generally note open bounds with parenthesis while E.U would use outward facing square brackets.
i.e. $(a,b) \text{ vs } ]a,b[$ or $[0,+\infty)\text{ vs }[0,+\infty[$
Best Answer
This is common notation when evaluating integrals.
$$ [f(t)]_{t_0}^{t_1} = f(t_1) - f(t_0), $$ though you usually need to deduce from the context what the variable is, it won’t always be $t$. (It’s $\theta$ in your case.)
Sometimes you will also see $$ [f(t)]_{t=t_0}^{t_1} $$ instead, which explicitly specifies the variable. Another common way of denoting the same thing is $$ f(t) \Big|_{t_0}^{t_1}, $$ again, sometimes with “$t = t_0$” in the subscript to disambiguate the variable.