Search results
Results From The WOW.Com Content Network
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 vector T(s) is given by
The total curvature of a closed curve is always an integer multiple of 2 π, where N is called the index of the curve or turning number – it is the winding number of the unit tangent vector about the origin, or equivalently the degree of the map to the unit circle assigning to each point of the curve, the unit velocity vector at that point.
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 gradient of a function is obtained by raising the index of the differential , whose components are given by: =; =; =, = = The divergence of a vector field with components is
In this definition, one says that a tangent vector to S at p is an assignment, to each local parametrization f : V → S with p ∈ f(V), of two numbers X 1 and X 2, such that for any other local parametrization f ′ : V → S with p ∈ f(V) (and with corresponding numbers (X ′) 1 and (X ′) 2), one has
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 ...
Another method of deriving vector and tensor derivative identities is to replace all occurrences of a vector in an algebraic identity by the del operator, provided that no variable occurs both inside and outside the scope of an operator or both inside the scope of one operator in a term and outside the scope of another operator in the same term ...
This page was last edited on 2 December 2024, at 16:34 (UTC).; Text is available under the Creative Commons Attribution-ShareAlike 4.0 License; additional terms may apply.