View source for Tangent space

A linear (more rarely, affine) space of [[tangent vector|vectors tangent]] to a [[smooth 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 can be identified with the linear space of derivations $D$ 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]].