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Here,

$$\left\lVert\frac{\partial\bf x}{\partial s}\times\frac{\partial\bf x}{\partial t}\right\rVert$$

the inside will at last be a vector. and two absolute value signs have covered it. what does it mean?

Can someone explain it to me? $||\vec a||$

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The length of the vector (also called the norm of the vector). – Chris Leary Mar 21 at 14:55
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up vote 8 down vote accepted

$\| a \|$ in general means the "norm" of $a$. Most commonly it means the Euclidean norm of the vector $a$. You could say "the geometric length of $a$" or "the magnitude of $a$" to refer to the same concept.

Be careful: there are many other norms that can be used to measure vectors, as well as norms that can be used to measure different sorts of objects entirely.

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the magnitude you mean? this? $\sqrt{a_x^2+a_y^2+a_z^2}$ – AHB Mar 21 at 14:57
    
@AmirBigdeli Amplitude is the wrong word, that refers to a size of oscillation. Magnitude is a correct word here, though. And yes, the Euclidean norm in 3 dimensions is $\sqrt{a_x^2+a_y^2+a_z^2}$. – Ian Mar 21 at 14:57

This generally indicate a norm in linear algebra and functional analysis. It can be thought as the length. For any $\mathbf{v} = [x,y,z]$ in $\mathbb{R}^3$, $\lVert \cdot \rVert$ represents the Euclidean norm $$ \lVert \mathbf{v} \rVert = \sqrt{x^2 + y^2 + z^2} $$

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In general, it can represent any norm you define. – Karlo Mar 21 at 22:30

it is the notation of norm (probably the euclidian one) https://en.wikipedia.org/wiki/Norm_(mathematics)

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Why isn't it like $|\vec a|$? I have seen this to refer to the magnitude of a vector. Or have I been wrong? – AHB Mar 21 at 15:01
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@AmirBigdeli It is sometimes not written as $|a|$ to avoid confusion with the absolute value of a number. People usually start writing it as $|a|$ again when norms of functions come into play, since it becomes more important to distinguish between norms of functions and norms of vectors than it is to distinguish between norms of vectors and absolute values of numbers. Also, the modulus of a complex number is exactly the same as the magnitude of the corresponding vector in $\mathbb{R}^2$, but that is always written as $|z|$, never $\| z \|$. – Ian Mar 21 at 15:05

This notation represents the magnitude of whatever is inside, generally a vector. See this.

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