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The product k 1 k 2 of the two principal curvatures is the Gaussian curvature, K, and the average (k 1 + k 2)/2 is the mean curvature, H. If at least one of the principal curvatures is zero at every point, then the Gaussian curvature will be 0 and the surface is a developable surface. For a minimal surface, the mean curvature is zero at every ...
The curvature is the norm of the derivative of T with respect to s. By using the above formula and the chain rule this derivative and its norm can be expressed in terms of γ′ and γ″ only, with the arc-length parameter s completely eliminated, giving the above formulas for the curvature.
Condition numbers can also be defined for nonlinear functions, and can be computed using calculus.The condition number varies with the point; in some cases one can use the maximum (or supremum) condition number over the domain of the function or domain of the question as an overall condition number, while in other cases the condition number at a particular point is of more interest.
Newton's method uses curvature information (i.e. the second derivative) to take a more direct route. In calculus , Newton's method (also called Newton–Raphson ) is an iterative method for finding the roots of a differentiable function f {\displaystyle f} , which are solutions to the equation f ( x ) = 0 {\displaystyle f(x)=0} .
Polynomial curves fitting points generated with a sine function. The black dotted line is the "true" data, the red line is a first degree polynomial, the green line is second degree, the orange line is third degree and the blue line is fourth degree. The first degree polynomial equation = + is a line with slope a. A line will connect any two ...
Furthermore, a surface which evolves under the mean curvature of the surface , is said to obey a heat-type equation called the mean curvature flow equation. The sphere is the only embedded surface of constant positive mean curvature without boundary or singularities. However, the result is not true when the condition "embedded surface" is ...
The determinant of the Hessian matrix, when evaluated at a critical point of a function, is equal to the Gaussian curvature of the function considered as a manifold. The eigenvalues of the Hessian at that point are the principal curvatures of the function, and the eigenvectors are the principal directions of curvature.
In differential geometry, the Gaussian curvature or Gauss curvature Κ of a smooth surface in three-dimensional space at a point is the product of the principal curvatures, κ 1 and κ 2, at the given point: =.