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.
Read more about Cotangent Space: Properties, The Differential of A Function, The Pullback of A Smooth Map, Exterior Powers
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—Max J. Friedländer (1867–1958)