The negative derivative

(1)

of the unit normal
vector field of a surface is called the shape operator
(or Weingarten map or second fundamental tensor). The shape operator is an extrinsic curvature,
and the Gaussian curvature is given by the
determinant of . If is a regular patch,
then
At each point
on a regular surface , the shape operator is a linear map

(4)

The shape operator for a surface is given by the Weingarten
equations.
See also
Curvature,
Fundamental
Forms,
Weingarten Equations
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References
Gray, A. "The Shape Operator," "Calculation of the Shape Operator," and "The Eigenvalues of the Shape Operator."
§16.1, 16.3, and 16.4 in Modern
Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca
Raton, FL: CRC Press, pp. 360363 and 367372, 1997.Reckziegel,
H. In Mathematical
Models from the Collections of Universities and Museums (Ed. G. Fischer).
Braunschweig, Germany: Vieweg, p. 30, 1986.Referenced on WolframAlpha
Shape Operator
Cite this as:
Weisstein, Eric W. "Shape Operator." From
MathWorldA Wolfram Web Resource. https://mathworld.wolfram.com/ShapeOperator.html
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