# Cotangent Space

In differential geometry, one can attach to every point x of a smooth (or differentiable) manifold a vector space called the cotangent space at x. Typically, the cotangent space is defined as the dual space of the tangent space at x, although there are more direct definitions (see below). The elements of the cotangent space are called cotangent vectors or tangent covectors.

### Other articles related to "space, cotangent space, cotangent, cotangent spaces, spaces":

Phase Space
... In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the ... For mechanical systems, the phase space usually consists of all possible values of position and momentum variables i.e ... the cotangent space of configuration space ...
Cotangent Space - Exterior Powers
... The k-th exterior power of the cotangent space, denoted Λk(Tx*M), is another important object in differential geometry ... of the k-th exterior power of the cotangent bundle, are called differential k-forms ...
Mathematics Of General Relativity - Tensors in General Relativity
... At each point of a manifold, the tangent and cotangent spaces to the manifold at that point may be constructed ... sometimes referred to as contravariant vectors) are defined as elements of the tangent space and covectors (sometimes termed covariant vectors, but more commonly ... At, these two vector spaces may be used to construct type tensors, which are real-valued multilinear maps acting on the direct sum of copies of the cotangent space with copies of the tangent space ...

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