= Mean curvature vector
{title2=$\mathbf H=\operatorname{tr}A$}
For an $n$-dimensional <embedded submanifold>, let $A$ denote its vector-valued <second fundamental form> and $e_1,\ldots,e_n$ an <orthonormal basis> of its <tangent space>. This branch uses the trace <mean curvature vector>
$$
\mathbf H=\sum_{i=1}^n A(e_i,e_i).
$$
An alternative averaged <mean curvature vector> is $\mathbf H/n$; on a <smooth surface> in three-dimensional <Euclidean space>, these are $2HN$ and $HN$, respectively. Reversing $N$ reverses $H$, so their product is independent of the chosen <orientation of a surface>. With the trace convention, the <first variation of area formula> has no additional factor $n$. For an outward-oriented <sphere> of <radius> $R$, $\mathbf H=-2N/R$: it points toward the centre even though the scalar's sign depends on conventions.
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