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The tangent space and the cotangent space at a point are both real vector spaces of the same dimension and therefore isomorphic to each other via many possible isomorphisms. The introduction of a Riemannian metric or a symplectic form gives rise to a natural isomorphism between the tangent space and the cotangent space at a point, associating ...
denote the tangent bundle and cotangent bundle, respectively, of the smooth manifold . , denote the tangent spaces of , at the points , , respectively. denotes the cotangent space of at the point .
The cotangent, or four-part, formulae relate two sides and two angles forming four consecutive parts around the triangle, for example (aCbA) or BaCb). In such a set there are inner and outer parts: for example in the set (BaCb) the inner angle is C, the inner side is a, the outer angle is B, the outer side is b.
By definition, an affine connection is a bilinear map () (), where () is a space of all vector fields on the spacetime. This bilinear map can be described in terms of a set of connection coefficients (also known as Christoffel symbols ) specifying what happens to components of basis vectors under infinitesimal parallel transport: ∇ e i e j ...
The other four trigonometric functions (tan, cot, sec, csc) can be defined as quotients and reciprocals of sin and cos, except where zero occurs in the denominator. It can be proved, for real arguments, that these definitions coincide with elementary geometric definitions if the argument is regarded as an angle in radians. [5]
cot – cotangent function. (Also written as ctg.) coth – hyperbolic cotangent function. cov – covariance of a pair of random variables. cover – coversine function. (Also written as covers, cvs, cosiv.) covercos – covercosine function. (Also written as cvc.) covers – coversine function. (Also written as cover, cvs, cosiv.) crd ...
For example, this is a way to describe the phase space of a pendulum. The state of the pendulum is determined by its position (an angle) and its momentum (or equivalently, its velocity, since its mass is constant). The entire state space looks like a cylinder, which is the cotangent bundle of the circle.
The dual space of a vector space is the set of real valued linear functions on the vector space. The cotangent space at a point is the dual of the tangent space at that point and the elements are referred to as cotangent vectors; the cotangent bundle is the collection of all cotangent vectors, along with the natural differentiable manifold ...