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A space curve; the vectors T, N, B; and the osculating plane spanned by T and N. In differential geometry, the Frenet–Serret formulas describe the kinematic properties of a particle moving along a differentiable curve in three-dimensional Euclidean space, or the geometric properties of the curve itself irrespective of any motion.
The dotted vector, in this case B, is differentiated, while the (undotted) A is held constant. The utility of the Feynman subscript notation lies in its use in the derivation of vector and tensor derivative identities, as in the following example which uses the algebraic identity C⋅(A×B) = (C×A)⋅B:
If t = s is the natural parameter, then the tangent vector has unit length. The formula simplifies: = ′ (). The unit tangent vector determines the orientation of the curve, or the forward direction, corresponding to the increasing values of the parameter.
Dividing this vector by its length yields a unit normal vector to the parametrized surface at a regular point: ^ = | |. In general, there are two choices of the unit normal vector to a surface at a given point, but for a regular parametrized surface, the preceding formula consistently picks one of them, and thus determines an orientation of the ...
The Frenet–Serret formulas show that there is a pair of functions defined on the curve, the torsion and curvature, which are obtained by differentiating the frame, and which describe completely how the frame evolves in time along the curve. A key feature of the general method is that a preferred moving frame, provided it can be found, gives a ...
Fix a curve : [,] with () = and () =. to parallel transport a vector to a vector in along , first extend to a vector field parallel along , and then take the value of this vector field at . The images below show parallel transport induced by the Levi-Civita connection associated to two different Riemannian metrics on the punctured plane R 2 ∖ ...
ρ is the length of the vector projected onto the xy-plane, φ is the angle between the projection of the vector onto the xy-plane (i.e. ρ) and the positive x-axis (0 ≤ φ < 2π), z is the regular z-coordinate. (ρ, φ, z) is given in Cartesian coordinates by:
The coordinate-independent definition of the square of the line element ds in an n-dimensional Riemannian or Pseudo Riemannian manifold (in physics usually a Lorentzian manifold) is the "square of the length" of an infinitesimal displacement [2] (in pseudo Riemannian manifolds possibly negative) whose square root should be used for computing curve length: = = (,) where g is the metric tensor ...