= Mean curvature
{title2=$H$}
{wiki}
<Mean curvature> measures the average signed bending of a <smooth surface>. Unlike <Gaussian curvature>, it adds the two <principal curvatures> rather than multiplying them. A <circular cylinder> therefore has nonzero <mean curvature> despite zero <Gaussian curvature>, whereas a <catenoid> has zero <mean curvature> despite negative <Gaussian curvature>. The cancellation on a <catenoid> explains how a curved <minimal surface> can be stationary for <surface area>.
Choose a <unit normal> $N$ and the <shape operator> $S=-dN$, as in the <second fundamental form> convention $II(v,w)=\langle Sv,w\rangle$. For a <smooth surface> in <Euclidean space> of dimension three, define the signed scalar <mean curvature> by
$$
H=\frac{k_1+k_2}{2}=\frac12\operatorname{tr}S=\frac{eG-2fF+gE}{2(EG-F^2)}.
$$
Here $E,F,G$ are the coefficients of the <first fundamental form>, and $e,f,g$ those of the <second fundamental form>. The factor $1/2$ makes $H$ an average; the scalar trace convention instead calls $k_1+k_2=2H$ the <mean curvature>. Authors who use $S=+dN$ reverse both signs. A formula involving <mean curvature> is therefore incomplete unless its <shape operator>, <unit normal>, and averaging convention are specified.
In <orientation reversal of surface curvature>, both <principal curvatures> and $H$ change sign, while <Gaussian curvature> and the <mean curvature vector> do not. The units of <mean curvature> are inverse <length>, compared with inverse squared <length> for <Gaussian curvature>. The distinction matters in <soap film pressure balance>: <pressure> couples to the sum of the signed <principal curvatures>.
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