2010 Mathematics Subject Classification: Primary: 58A05 Secondary: 58A32 [MSN][ZBL]
A linear (more rarely, affine) space of vectors tangent to a smooth (differentiable) manifold (curve, surface, $\dots$) $M$ at a given point $a\in M$. One of the standard notations is $T_aM$.
For curves, surfaces and submanifolds embedded in a Euclidean subspace $\R^n$ the tangent subspace can be identified with an affine subset (of the corresponding dimension $1,2,\dots$) in the ambient space, passing through $a$. For abstract manifolds $T_aM$ is naturally isomorphic to the linear space of derivations $D:C^\infty(M)\to\R$ of the ring of smooth functions on $M$ satisfying the Leibniz rule: $$ D\in T_a M\iff D:C^\infty(M)\to\R,\qquad D(f\pm g)=Df\pm Dg,\ D(\lambda f)=\lambda Df,\ D(f\cdot g)=f(a)\cdot Dg+g(a)\cdot Df. $$ The (disjoint) union of tangent spaces to a manifold to all its points has a natural structure of a bundle over $M$, called the tangent bundle.
For submanifolds of $\R^n$ the tangent space $T_aM$ coincides with the tangent cone defined as the union of limits of secants.
Tangent space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tangent_space&oldid=30964