Definition & Betydelse | Engelska ordet COTANGENT

Definition av COTANGENT

1. cotangens

Exempel på hur man kan använda COTANGENT i en mening

• Vectors (sometimes referred to as contravariant vectors) are defined as elements of the tangent space and covectors (sometimes termed covariant vectors, but more commonly dual vectors or one-forms) are elements of the cotangent space.
• Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios.
• where "arsinh" is the inverse hyperbolic sine function and "cot" is the (circular) cotangent function.
• The prefix "co-" (in "cosine", "cotangent", "cosecant") is found in Edmund Gunter's Canon triangulorum (1620), which defines the cosinus as an abbreviation for the sinus complementi (sine of the complementary angle) and proceeds to define the cotangens similarly.
• In the work of Lenhard Ng, relative SFT is used to obtain invariants of smooth knots: a knot or link inside a topological three-manifold gives rise to a Legendrian torus inside a contact five-manifold, consisisting of the unit conormal bundle to the knot inside the unit cotangent bundle of the ambient three-manifold.
• Just as we build differential forms out of exterior powers of the cotangent bundle, we can build exterior powers of the complexified cotangent bundle (which is canonically isomorphic to the bundle of dual spaces of the complexified tangent bundle).
• On the other hand, the ring of germs of smooth functions at a point in an n-manifold has an n-dimensional Zariski cotangent space.
• In differential geometry, a one-form (or covector field) on a differentiable manifold is a differential form of degree one, that is, a smooth section of the cotangent bundle.
• Trigonometric functions: sine, cosine, tangent, cotangent, secant, cosecant, exsecant, excosecant, versine, coversine, vercosine, covercosine, haversine, hacoversine, havercosine, hacovercosine, Inverse trigonometric functions etc.
• The angle opposite the leg of length 1 (this angle can be labeled φ = π/2 − θ) has cotangent equal to the length of the other leg, and cosecant equal to the length of the hypotenuse.
• This property characterizes spinors: spinors can be viewed as the "square roots" of vectors (although this is inaccurate and may be misleading; they are better viewed as "square roots" of sections of vector bundles – in the case of the exterior algebra bundle of the cotangent bundle, they thus become "square roots" of differential forms).
• The strongest results are obtained for over-determined systems (holonomic systems), and on the characteristic variety cut out by the symbols, which in the good case is a Lagrangian submanifold of the cotangent bundle of maximal dimension (involutive systems).
• For example, in general relativity, the gravitational field is described through the metric tensor, which is a tensor field with one tensor at each point of the space-time manifold, and each belonging to the tensor product of the cotangent space at the point with itself.
• Modern mathematical descriptions of canonical transformations are considered under the broader topic of symplectomorphism which covers the subject with advanced mathematical prerequisites such as cotangent bundles, exterior derivatives and symplectic manifolds.
• the hyperbolic sine and cosine functions (but not hyperbolic tangent, cotangent, secant, or cosecant).
• The same is true of secant (Latin: secans) and cosecant (Latin: cosecans, secans complementi) as well as of tangent (Latin: tangens) and cotangent (Latin: cotangens, tangens complementi):.
• In the late 1960s, Michel André and Daniel Quillen independently came up with the correct definition for a morphism of commutative rings, using simplicial methods to make precise the idea of the cotangent complex as given by taking the (non-abelian) left derived functor of Kähler differentials.
• Examples of symplectomorphisms include the canonical transformations of classical mechanics and theoretical physics, the flow associated to any Hamiltonian function, the map on cotangent bundles induced by any diffeomorphism of manifolds, and the coadjoint action of an element of a Lie group on a coadjoint orbit.
• All cotangent spaces at points on a connected manifold have the same dimension, equal to the dimension of the manifold.
• There are six in common use: inverse hyperbolic sine, inverse hyperbolic cosine, inverse hyperbolic tangent, inverse hyperbolic cosecant, inverse hyperbolic secant, and inverse hyperbolic cotangent.

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