Let be a point of a surface
where has the continuous first and partial derivatives in a neighbourhood of . If one intersects the surface with a plane containing the surface normal at , the intersection curve is called a normal section.
When the direction of the intersecting plane is varied, one gets different normal sections, and their curvatures (http://planetmath.org/CurvaturePlaneCurve) at , the so-called normal curvatures, vary having a minimum value and a maximum value . The arithmetic mean of and is called the mean curvature of the surface at .
By the suppositions on the function , examining the normal curvatures can without loss of generality be to the following: Examine the curvature of the normal sections through the origin, the surface given in the form
where has the continuous first and partial derivatives in a neighbourhood of the origin and
Indeed, one can take a new rectangular coordinate system with the new origin and the normal at the new -axis; then the new -plane coincides with the tangent plane of the surface (1) at . The equation (1) defines the function of (2).
|Date of creation||2013-03-22 17:26:25|
|Last modified on||2013-03-22 17:26:25|
|Last modified by||pahio (2872)|