An extrinsic scalar measure of how a surface bends in ambient space, defined at a point as the average of the two principal curvatures, commonly H = (k1 + k2)/2; it equals half the trace of the shape operator.

Principle

Mean curvature arises as the first variation of surface area: its vanishing characterizes stationary points of the area functional under normal deformations; sign depends on orientation convention.

Demonstration

A soap film spanning a wire frame forms a surface with mean curvature H≈0 (a minimal surface); a round sphere of radius R has H = 1/R pointing toward the center for the outward normal convention.

Misapplication

Interpreting mean curvature as an intrinsic invariant of the surface metric or applying H at points without a chosen unit normal (non-orientable surfaces) without specifying orientation.

Consequence

Surfaces with H=0 minimize area to first order and obey elliptic PDEs for graphical parametrizations; prescribed-mean-curvature problems produce physically relevant shapes (e.g., capillarity interfaces).

Reversal

Gaussian curvature multiplies principal curvatures and is intrinsic; replacing average by product changes invariance and geometric interpretation drastically.

Boundary

Defined where the surface is at least C^2 and an orientation (unit normal) is chosen; sign and existence fail at singular points, for non-orientable surfaces without local orientation, or for distributions lacking second derivatives.

Semantic Tension

Tension exists between mean curvature interpreted as an extrinsic bending measure and its variational role as Euler–Lagrange for area; sign conventions and vector-valued mean curvature (curvature vector) are common sources of ambiguity.

Synthesis

Mean curvature is the oriented average of principal curvatures, an extrinsic curvature scalar that controls first-order area variation and appears in geometric PDEs describing minimal and capillary surfaces.