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Geometric relevance: The torsion τ(s) measures the turnaround of the binormal vector. The larger the torsion is, the faster the binormal vector rotates around the axis given by the tangent vector (see graphical illustrations). In the animated figure the rotation of the binormal vector is clearly visible at the peaks of the torsion function.
The torsion tensor is a bilinear map of two input vectors ,, that produces an output vector (,) representing the displacement within a tangent space when the tangent space is developed (or "rolled") along an infinitesimal parallelogram whose sides are ,.
The first Frenet-Serret formula holds by the definition of the normal N and the curvature κ, and the third Frenet-Serret formula holds by the definition of the torsion τ. Thus what is needed is to show the second Frenet-Serret formula. Since T, N, B are orthogonal unit vectors with B = T × N, one also has T = N × B and N = B × T.
Torsion of a square section bar Example of torsion mechanics. In the field of solid mechanics, torsion is the twisting of an object due to an applied torque. [1] [2] Torsion could be defined as strain [3] [4] or angular deformation, [5] and is measured by the angle a chosen section is rotated from its equilibrium position. [6]
the tangent vector at point p of the i th coordinate curve. The ∂ i are a natural basis for the tangent space at point p, and the X i the corresponding coordinates for the vector field X = X i ∂ i. When both curvature and torsion vanish, the connection defines a pre-Lie algebra structure on the space of global sections of the tangent bundle.
Then it follows immediately from the definitions that M is torsion-free (if the ring R is not a domain then torsion is considered with respect to the set S of non-zero-divisors of R). In particular, any free abelian group is torsion-free and any vector space over a field K is torsion-free when viewed as a module over K.
To describe the torsion, note that the vector bundle E is the tangent bundle. This carries a canonical solder form (sometimes called the canonical one-form , especially in the context of classical mechanics ) that is the section θ of Hom(T M , T M ) = T ∗ M ⊗ T M corresponding to the identity endomorphism of the tangent spaces.
is the force vector, F is the magnitude of the force vector and F ⊥ is the amount of force directed perpendicularly to the position of the particle, denotes the cross product, which produces a vector that is perpendicular both to r and to F following the right-hand rule,