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Encyclopedia > Torsion

# Mathmatics

In mathematics, the term torsion has several meanings, mostly unrelated to each other. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ...

## Differential geometry of curves GA_googleFillSlot("encyclopedia_square");

In the elementary differential geometry of curves in three dimensions, the torsion of a curve measures how sharply it is twisting. It is analogous to curvature in two dimensions. Given a function r(t) with values in R3, the torsion at a given value of t is In mathematics, the differential geometry of curves provides definitions and methods to analyze smooth curves in Riemannian manifolds and Pseudo-Riemannian manifolds (and in particular in Euclidean space) using differential and integral calculus. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... Curvature is the amount by which a geometric object deviates from being flat. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...

Here the primes denote the derivatives of r with respect to t; if the cross product in the denominator is zero, the torsion τ is defined to be zero as well. The derivative in mathematics (specifically, differential calculus) is a quantity that measures, on continuous functions, the limit of a rate of change, , as approaches 0. ... In mathematics, the cross product is a binary operation on vectors in vector space. ...

The torsion of a curve will be zero if and only if the curve sits inside a fixed plane. It is positive for right-handed spirals and negative for left-handed ones.

## Torsion tensor of a connection

A second meaning of torsion in differential geometry is the torsion tensor, which depends on an affine connection . It is a (1,2) tensor given by the formula In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... An affine connection is a connection on the tangent bundle of a differentiable manifold. ... In mathematics, a tensor is a certain kind of geometrical entity, or alternatively generalized quantity. The tensor concept includes the ideas of scalar, vector and linear operator. ...

where [u,v] is the Lie bracket of the two vector fields. In mathematics, a Lie derivative, named after Sophus Lie, is a derivation on the algebra of smooth functions over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by The Lie derivatives are represented...

A connection is torsion free if its torsion tensor is identically zero. Torsion free connections are considered most frequently - the Levi-Civita connection is assumed to have zero torsion, for instance. In Riemannian geometry, the Levi-Civita connection (named for Tullio Levi-Civita) is the torsion-free connection of the tangent bundle, preserving a given Riemannian metric (or pseudo-Riemannian metric). ...

See also Connection form. In mathematics, and specifically differential geometry, the connection form captures the invariant aspects of the connection on principal bundles, vector bundles and line bundles. ...

## Abstract algebra

In abstract algebra, the torsion subgroup of an abelian group consists of all elements of finite order. An abelian group is called torsion-free if and only if the identity is the only element that has finite order. (This concept generalises to that of a torsion module.) In the Tor functors of homological algebra, which arise because tensor product does not in general preserve exact sequences, the symbol Tor does stand for this kind of algebraic torsion, historically speaking anyway. These functors were introduced in order to make systematic the universal coefficient theorem of homology theory, in cases where the homology groups Hi(X,Z) of a space X had some torsion. Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... In group theory, the torsion subgroup of an abelian group A is the subgroup of A consisting of all elements that have finite order. ... In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... In group theory, the term order is used in two closely related senses: the order of a group is its cardinality, i. ... In abstract algebra, a branch of mathematics, a torsion module is a module which, in effect, ignores the action of its ring. ... The Tor functors are the derived functors of the tensor product functor in mathematics. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ... In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ... In mathematics, the universal coefficient theorem in algebraic topology establishes the relationship in homology theory between the integral homology of a topological space X, and its homology with coefficients in any abelian group A. It shows that the integral homology groups Hi(X,Z) do in a certain, definite sense... In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homeos = identical and logos = word) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ...

## Topology

Some topological invariants are called torsions: for example the Reidemeister-Schreier torsion of a group acting on a finite complex; and also the analytic torsion defined using Laplacians. In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. ...

In group theory, the torsion subgroup of an abelian group A is the subgroup of A consisting of all elements that have finite order. ...

# Mechanics

In solid mechanics, torsion is when an object is twisted or screwed around its axis. Torsion can be the result of an applied torque. It is a kind of shear stress. Solid mechanics (also known as the theory of elasticity) is a branch of physics, which governs the response of solid material to applied stress (e. ... The concept of torque in physics, also called moment or couple, originated with the work of Archimedes on levers. ... Shear stress is a stress state where the shape of a material tends to change (usually by sliding forces - torque by transversely-acting forces) without particular volume change. ...

A torsion coefficient is used to describe torsional springs. ... Simple Gravity Pendulum assumues no air resistance and no friction of/at the nail/screw. ... A torsion spring is a ribbon, bar, or coil that reacts against twisting motion. ... The concept of torque in physics, also called moment or couple, originated with the work of Archimedes on levers. ...

# Other

• Torsion is also sometimes used for the medical condition commonly known as bloat.
• Torsion fields are infamous as a pseudoscientific field popular in Russia.

Results from FactBites:

 Torsion - Wikipedia, the free encyclopedia (401 words) Torsion free connections are considered most frequently - the Levi-Civita connection is assumed to have zero torsion, for instance. Some topological invariants are called torsions: for example the Reidemeister-Schreier torsion of a group acting on a finite complex; and also the analytic torsion defined using Laplacians. Torsion is also sometimes used for the medical condition commonly known as bloat.
 Torsion beam suspension - Wikipedia, the free encyclopedia (332 words) Torsion beam suspension, also known as a torsion bar or torsion spring, is a vehicle suspension system. One end of a long metal bar is attached firmly to the vehicle chassis; the opposite end has a lever, perpendicular to the bar, that is attached to the axle of a wheel. Some vehicles use torsion bars to implement automatic leveling with sensors and motors to dynamically adjust the torsion in each bar in response to changing loads and road conditions.
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