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The Frenet–Serret formulas were generalized to higher-dimensional Euclidean spaces by Camille Jordan in 1874. Suppose that r(s) is a smooth curve in and that the first n derivatives of r are linearly independent. [2] The vectors in the Frenet–Serret frame are an orthonormal basis constructed by applying the Gram-Schmidt process to the ...
A normal plane at p is one that contains the normal vector, and will therefore also contain a unique direction tangent to the surface and cut the surface in a plane curve, called normal section. This curve will in general have different curvatures for different normal planes at p. The principal curvatures at p, denoted k 1 and k 2, are the ...
Gauss–Codazzi equations. Fundamental formulas linking the metric and curvature tensor of a manifold. In Riemannian geometry and pseudo-Riemannian geometry, the Gauss–Codazzi equations (also called the Gauss–Codazzi–Weingarten-Mainardi equations or Gauss–Peterson–Codazzi formulas[1]) are fundamental formulas that link together the ...
In geometry, a normal is an object (e.g. a line, ray, or vector) that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the line perpendicular to the tangent line to the curve at the point. A normal vector of length one is called a unit normal vector. A curvature vector is a normal vector ...
where n(t) is the "principal" unit normal to the curve in the surface, constructed by rotating the unit tangent vector ċ(t) through an angle of +90°. The geodesic curvature at a point is an intrinsic invariant depending only on the metric near the point.
Normal plane (geometry) Saddle surface with normal planes in directions of principal curvatures. In geometry, a normal plane is any plane containing the normal vector of a surface at a particular point. The normal plane also refers to the plane that is perpendicular to the tangent vector of a space curve; (this plane also contains the normal ...
At a point take a path with given tangent vector and then map it to the tangent vector of the lifted path. (For vector fields the curves can be taken to be the integral curves of a local flow.) In this way any vector field X on M can be lifted to a vector field X* on F satisfying [32] X* is a vector field on F; the map X ↦ X* is C ∞ (M)-linear;
Euler's second axiom or law (law of balance of angular momentum or balance of torques) states that in an inertial frame the time rate of change of angular momentum L of an arbitrary portion of a continuous body is equal to the total applied torque M acting on that portion, and it is expressed as. where is the velocity, the volume, and the ...
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