“tangent space”的概念、定义、习题解题方法及翻译-数学辞典

单词

tangent space

释义

tangent space

A tangent vector to a point P on a smooth curve is any vector parallel to the tangent line. If r(s) is a parameterization of the curve by arc length, then t = r´(s) is a unit tangent vector (see also Serret-Frenet formulae). More generally, a tangent vector to a point P in a smooth manifold X is a tangent vector to a smooth curve in the manifold passing through P.

The tangent space at P is the vector space whose elements are the tangent vectors at P. So, if r(u,v) is a parameterization of a surface about P = r(u0,v0), then the tangent space is spanned by ∂r/∂u(u0,v0) and ∂r/∂v(u0,v0). Note that the tangent space is parallel to the tangent plane but passes through the origin. The tangent space at P is denoted TP(X).

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