Are $\exp^{-a\|x\|^{2}}$ and $f(x) = \exp^{-a|x|^{2}}$ the same? What is $\|x\|$?

Question

I want to compute the integral of $f(x)$ using the Monte Carlo integration method.

$f(x) = \exp^{-a\|x\|^{2}}$ over the cube $[-1,1]^n$, where $a$ is any constant and $n$ dimension of space.

$[-1,1]^n$ means $(r_1,r_2,r_3,\dotsc, r_n)$ where $-1 < r_i < 1$.

I want to know does

$$f(x) = \exp^{-a\|x\|^2}$$ and $$f(x) = \exp^{-a|x|^{2}}$$ are same?

Here $x$ is a vector of $n$-dimensions (or point in $n$-dimensional space). So what is $|x|$ and $\|x\|$? Is it a determinant or matrix or vector or something else? I welcome your hint or pointer to hint. Thank you.

Answer 1

In this case, without any other context, I would assume that $$|x| = \|x\| = \sqrt{x_1^2 + x_2^2 + \dotsb + x_n^2}.$$