... inequality,¹

The triangle inequality, which is one of the defining properties of a norm, states that

$$\Vert x+y\Vert\le\Vert x\Vert+\Vert y\Vert\ \mbox{for all}\x,y\in{\bf {\rm R}}^n.$$

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... inequality,²

The reverse triangle inequality, which can be proved from the triangle inequality, states that

$$\Vert x-y\Vert\ge\left\vert\,\Vert x\Vert-\Vert y\Vert\,\right\vert\ \mbox{for all}\x,y\in{\bf {\rm R}}^n.$$

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... integrand.³

A digression: Equation (4) is commonly used when one wants to use a Mean Value Theorem (MVT). For a function $f:{\bf {\rm R}}\rightarrow{\bf {\rm R}}$ of one variable, the MVT (a special case of Taylor's theorem) states that if f is sufficiently smooth, then there exists $c\in (a,b)$ such that

f(b)=f(a)+f'(c)(b-a).

However, the MVT does not hold for vector-valued functions, because the number c is typically different for each component F_i. Equation (4) is usually an adequate substitute.

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