At a regular point of a smooth surface, the arithmetic mean H=(k1+k2)/2 of the two principal curvatures; an extrinsic measure of local bending related to the surface's normal variation.

Principle

Mean curvature arises as the first variation of surface area: to first order, displacing a surface along its normal changes area proportionally to H, so H=0 characterizes stationary-area (minimal) surfaces.

Demonstration

A round sphere of radius R has k1=k2=1/R so H=1/R; a minimal surface such as a catenoid or a soap film patch satisfies H=0 at every point.

Misapplication

Using mean curvature as an intrinsic invariant or confusing H with Gaussian curvature; ignoring orientation sign which makes H change sign when the unit normal is reversed.

Consequence

Mean curvature governs physical equilibria of interfaces (soap films, capillarity) and drives geometric flows (mean curvature flow) that smooth shapes toward lower-area configurations.

Reversal

Replacing averaging by multiplication yields Gaussian curvature; switching to normal curvature along a direction extracts directional bending rather than the averaged normal variation.

Boundary

Defined for smooth regular surfaces with a chosen unit normal; undefined at singularities or nonorientable surfaces without a consistent normal. Sign conventions vary between authors.

Semantic Tension

Tension exists between extrinsic significance (depends on embedding and chosen normal) and its variational role (intrinsic functional derivative of area); one must distinguish geometric role from physical interpretation.

Synthesis

Mean curvature is the average of the principal curvatures at a point, an extrinsic scalar measuring normal bending that dictates area variation, physical equilibrium shapes, and smoothing dynamics.