Let p be a point on the surface S inside the three dimensional Euclidean space . Each plane through p containing the normal line to S cuts S in a (plane) curve. Fixing a choice of unit normal gives a signed curvature to that curve. As the plane is rotated by an angle \theta (always containing the normal line) that curvature can vary. The
maximal curvature \kappa_1 and
minimal curvature \kappa_2 are known as the
principal curvatures of S. The
mean curvature at p\in S is then the average of the signed curvature over all angles \theta: :H = \frac{1}{2\pi}\int_0^{2\pi} \kappa(\theta) \;d\theta. By applying
Euler's theorem, this is equal to the average of the principal curvatures : :H = {1 \over 2} (\kappa_1 + \kappa_2). More generally , for a
hypersurface T the mean curvature is given as :H=\frac{1}{n}\sum_{i=1}^{n} \kappa_{i}. More abstractly, the mean curvature is the trace of the
second fundamental form divided by
n (or equivalently, the
shape operator). Additionally, the mean curvature H may be written in terms of the
covariant derivative \nabla as :H\vec{n} = g^{ij}\nabla_i\nabla_j X, using the
Gauss-Weingarten relations, where X(x) is a smoothly embedded hypersurface, \vec{n} a unit normal vector, and g_{ij} the
metric tensor. A surface is a
minimal surface if and only if the mean curvature is zero. Furthermore, a surface which evolves under the mean curvature of the surface S, is said to obey a
heat-type equation called the
mean curvature flow equation. The
sphere is the only embedded surface of constant positive mean curvature without boundary or singularities. However, the result is not true when the condition "embedded surface" is weakened to "immersed surface".
Surfaces in 3D space For a surface defined in 3D space, the mean curvature is related to the
divergence of a unit
normal of the surface: :2 H = -\nabla \cdot \hat n where the sign of the curvature depends on the choice of normal (inward or outward): the curvature is positive if the surface curves "towards" the normal. The formula above holds for surfaces in 3D space defined in any manner, as long as the divergence of the unit normal may be calculated. Mean curvature may also be calculated as : 2 H = \text{Trace}((\mathrm{II})(\mathrm{I}^{-1})) where I and II denote
first and
second fundamental forms, respectively. If S(x,y) is a parametrization of the surface and u, v are two linearly independent vectors in
parameter space then the mean curvature can be written in terms of the first and second quadratic form matrices as \frac{l G-2 m F + n E}{2 ( E G - F^2)} where E = \mathrm{I}(u,u), F = \mathrm{I}(u,v), G = \mathrm{I}(v,v), l = \mathrm{II}(u,u), m = \mathrm{II}(u,v), n = \mathrm{II}(v,v). For the special case of a surface defined as a function of two coordinates (a
bivariate function), e.g. z = S(x, y), and using the upward pointing normal, the (doubled) mean curvature expression is :\begin{align} 2 H & = -\nabla \cdot \left(\frac{\nabla(z-S)}\right) \\ & = \nabla \cdot \left(\frac{\nabla S-\nabla z} {\sqrt{1 + |\nabla S|^2}}\right) \\ & = \frac{ \left(1 + \left(\frac{\partial S}{\partial x}\right)^2\right) \frac{\partial^2 S}{\partial y^2} - 2 \frac{\partial S}{\partial x} \frac{\partial S}{\partial y} \frac{\partial^2 S}{\partial x \partial y} + \left(1 + \left(\frac{\partial S}{\partial y}\right)^2\right) \frac{\partial^2 S}{\partial x^2} }{\left(1 + \left(\frac{\partial S}{\partial x}\right)^2 + \left(\frac{\partial S}{\partial y}\right)^2\right)^{3/2}}. \end{align} In particular at a point where \nabla S=0, the mean curvature is half the trace of the Hessian matrix of S. If the surface is additionally known to be
axisymmetric with z = S(r), :2 H = \frac{\frac{\partial^2 S}{\partial r^2}}{\left(1 + \left(\frac{\partial S}{\partial r}\right)^2\right)^{3/2}} + {\frac{\partial S}{\partial r}}\frac{1}{r \left(1 + \left(\frac{\partial S}{\partial r}\right)^2\right)^{1/2}}, where {\frac{\partial S}{\partial r}} \frac{1}{r} comes from the derivative of z = S(r) = S\left(\sqrt{x^2 + y^2} \right).
Implicit form of mean curvature The mean curvature of a surface specified by an equation F(x,y,z)=0 can be calculated by using the gradient \nabla F=\left( \frac{\partial F}{\partial x}, \frac{\partial F}{\partial y}, \frac{\partial F}{\partial z} \right) and the
Hessian matrix :\textstyle \mbox{Hess}(F)= \begin{pmatrix} \frac{\partial^2 F}{\partial x^2} & \frac{\partial^2 F}{\partial x\partial y} & \frac{\partial^2 F}{\partial x\partial z} \\ \frac{\partial^2 F}{\partial y\partial x} & \frac{\partial^2 F}{\partial y^2} & \frac{\partial^2 F}{\partial y\partial z} \\ \frac{\partial^2 F}{\partial z\partial x} & \frac{\partial^2 F}{\partial z\partial y} & \frac{\partial^2 F}{\partial z^2} \end{pmatrix} . The mean curvature is given by: :H = \frac{ \nabla F\ \mbox{Hess}(F) \ \nabla F^{\mathsf {T}} - |\nabla F|^2\, \text{Trace}(\mbox{Hess}(F)) } { 2|\nabla F|^3 } Another form is as the
divergence of the unit normal. A unit normal is given by \frac{\nabla F} and the mean curvature is :H = -{\frac{1}{2}}\nabla\cdot \left(\frac{\nabla F}\right). ==In fluid mechanics==