A tangent vector to a smooth curve is the derivative of a parametrization of that curve. Tangent vectors to a surface form its tangent plane.
At on a smooth manifold, a derivation at that point is a real-linear map on germs of smooth functions with . It defines a tangent vector. The local factorization gives , so the coordinate derivations form a basis of the tangent space.
The tangent space is the vector space of tangent vectors to a smooth manifold at . For an embedded manifold , it is
(1)
For a regular parametrized surface , it is spanned by and .
A tangent vector is a linear map on germs of smooth functions satisfying . In a manifold chart, first-order expansion shows . Thus the coordinate derivations form a basis, and the dimension equals the manifold dimension. Composition of germs with a smooth map defines its differential of a smooth map without choosing charts.
The cotangent space at is the dual vector space . Its elements are covectors on the tangent space. In a manifold chart , the basis is dual to . Under a coordinate change to , the coefficients of obey . The disjoint union of these spaces is the cotangent bundle.
A Riemannian metric induces a dual inner product on each cotangent space. On decomposable covectors define and extend bilinearly. Increasing wedges of an orthonormal coframe form an orthonormal basis. This positive-definite inner product is the one used to define the Hodge star operator.
If is a local parametrization with , then
(1)
The derivative has rank , so this is a -dimensional vector subspace. A change of parametrization composes with an invertible coordinate-change derivative, leaving its image unchanged.
For an embedded smooth manifold , if , , and , then . In a local graph representation , write and use continuity of .
Local coordinates induce the coordinate-basis tangent vectors . Under a coordinate change they transform by the chain rule.
The unit tangent vector of a regular parametrized curve is
(1)
The velocity vector of a parametrized curve is its tangent vector .

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Tangent vector by Wikipedia Bot 1
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A tangent vector is a mathematical concept from differential geometry and calculus that describes a vector that is tangent to a curve or surface at a certain point. Here are some key points about tangent vectors: 1. **Geometric Interpretation**: At any given point on a curve in a multidimensional space, the tangent vector represents the direction in which the curve is moving at that point.
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