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Animation of the curvature and the acceleration vector T′(s) As in the case of curves in two dimensions, the curvature of a regular space curve C in three dimensions (and higher) is the magnitude of the acceleration of a particle moving with unit speed along a curve. Thus if γ(s) is the arc-length parametrization of C then the unit tangent ...
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} .
Indeed, if is a vector of unit length on a Riemannian -manifold, then (,) is precisely () times the average value of the sectional curvature, taken over all the 2-planes containing . There is an ( n − 2 ) {\displaystyle (n-2)} -dimensional family of such 2-planes, and so only in dimensions 2 and 3 does the Ricci tensor determine the full ...
A tangential vector field X on S assigns, to each p in S, a tangent vector X p to S at p. According to the "intrinsic" definition of tangent vectors given above, a tangential vector field X then assigns, to each local parametrization f : V → S, two real-valued functions X 1 and X 2 on V, so that
Radius of curvature and center of curvature. In differential geometry, the radius of curvature, R, is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius of a circle that best fits a normal section or ...
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 Berry curvature per solid angle is given by ¯ = / = /. In this case, the Berry phase corresponding to any given path on the unit sphere S 2 {\displaystyle {\mathcal {S}}^{2}} in magnetic-field space is just half the solid angle subtended by the path.
The equivalence classes are called C r-curves and are central objects studied in the differential geometry of curves. Two parametric C r-curves, : and :, are said to be equivalent if and only if there exists a bijective C r-map φ : I 1 → I 2 such that : ′ and : (()) = (). γ 2 is then said to be a re-parametrization of γ 1.