# Mean curvature

of a surface $\Phi ^ {2}$ in $3$-dimensional Euclidean space $\mathbf R ^ {3}$

Half of the sum of the principal curvatures (cf. Principal curvature) $k _ {1}$ and $k _ {2}$, calculated at a point $A$ of this surface:

$$H( A) = \frac{k _ {1} + k _ {2} }{2} .$$

For a hypersurface $\Phi ^ {n}$ in the Euclidean space $\mathbf R ^ {n+1}$, this formula is generalized in the following way:

$$H( A) = \frac{k _ {1} + \dots + k _ {n} }{n} ,$$

where $k _ {i}$, $i = 1 \dots n$, are the principal curvatures of the hypersurface, calculated at a point $A \in \Phi ^ {n}$.

The mean curvature of a surface in $\mathbf R ^ {3}$ can be expressed by means of the coefficients of the first and second fundamental forms of this surface:

$$H( A) = \frac{1}{2} \frac{LG - 2MF + NE }{EG - F ^ { 2 } } ,$$

where $E, F, G$ are the coefficients of the first fundamental form, and $L, M, N$ are the coefficients of the second fundamental form, calculated at a point $A \in \Phi ^ {2}$. In the particular case where the surface is defined by an equation $z = f( x, y)$, the mean curvature is calculated using the formula:

$$H ( A) =$$

$$= \ \frac{\left ( 1 + \left ( \frac{\partial f }{\partial y } \right ) ^ {2} \right ) \frac{\partial ^ {2} f }{\partial x ^ {2} } - 2 \frac{\partial f }{\partial x } \frac{\partial f }{\partial y } \frac{\partial ^ {2} f }{\partial x \partial y } + \left ( 1 + \left ( \frac{\partial f }{ \partial x } \right ) ^ {2} \right ) \frac{\partial ^ {2} f }{\partial y ^ {2} } }{\left ( 1 + \left ( \frac{\partial f }{\partial x } \right ) ^ {2} + \left ( \frac{\partial f }{\partial y } \right ) ^ {2} \right ) ^ {3/2} } ,$$

which is generalized for a hypersurface $\Phi ^ {n}$ in $\mathbf R ^ {n+1}$, defined by the equation $x _ {n+1} = f( x _ {1} \dots x _ {n} )$, as follows:

$$H ( A) = \frac{\sum _ { i= 1} ^ { n } \left ( 1 + p ^ {2} - \left ( \frac{\partial f }{\partial x _ {i} } \right ) ^ {2} \right ) \frac{\partial ^ {2} f }{\partial x _ {i} ^ {2} } - \sum _ { i,j= 1} ^ { n } \frac{\partial f }{\partial x _ {i} } \frac{\partial f }{\partial x _ {j} } \frac{ \partial ^ {2} f }{\partial x _ {i} \partial x _ {j} } }{( 1 + p ^ {2} ) ^ {3/2} } ,$$

where

$$p ^ {2} = | \mathop{\rm grad} f | ^ {2} = \ \left ( \frac{\partial f }{\partial x _ {1} } \right ) ^ {2} + \dots + \left ( \frac{\partial f }{\partial x _ {n} } \right ) ^ {2} .$$

For an $m$-dimensional submanifold $M$ of an $n$-dimensional Euclidean space of codimension $n - m > 1$, the mean curvature generalizes to the notion of the mean curvature normal
$$\nu _ {p} = \frac{1}{m} \sum _ { j= 1} ^ { n- m} [ \mathop{\rm Tr} A ( e _ {j} ) ] e _ {j} ,$$
where $e _ {1} \dots e _ {n-m}$ is an orthonormal frame of the normal space (cf. Normal space (to a surface)) of $M$ at $p$ and $A ( e _ {j} ) : T _ {p} M \rightarrow T _ {p} M$ ($T _ {p} M$ denotes the tangent space to $M$ at $p$) is the shape operator of $M$ at $p$ in the direction $e _ {j}$, which is related to the second fundamental tensor $V$ of $M$ at $p$ by $\langle A ( e _ {j} ) ( X) , Y \rangle = \langle V ( X , Y ) , e _ {j} \rangle$.