instance, to the varieties considered in algebraic geometry. If is a derivation at , then for every , which means that gives rise to a linear map . Conversely, if is a linear map, then defines a derivation at . This yields an equivalence between tangent spaces defined via derivations and tangent spaces defined via cotangent spaces. Properties If is an open subset of , then is a manifold in a natural manner (take coordinate charts to be identity maps on open subsets of ), and the tangent spaces are all naturally identified with . Tangent vectors as directional derivatives Another way to think about tangent vectors is as directional derivatives. Given a vector in , one defines the corresponding directional derivative at a point by This map is naturally a derivation at . Furthermore, every derivation at a point in is of this form. Hence, there is a one-to-one correspondence between vectors (thought of as tangent vectors at a point) and derivations at a point. As tangent vectors to a general manifold at a point can be defined as derivations at that point, it is natural to think of them as directional derivatives. Specifically, if is a tangent vector to at a point (thought of as a derivation), then define the directional derivative in the direction by If we think of as the initial velocity of a differentiable curve initialized at , i.e., , then instead, define by Basis of the tangent space at a point For a manifold , if a chart is given with , then one can define an ordered basis of by Then for every tangent vector , one has This formula therefore expresses as a linear combination of the basis tangent vectors defined by the coordinate chart . The derivative of a map Every smooth (or differentiable) map between smooth (or differentiable) manifolds induces natural linear maps between their corresponding tangent spaces: If the tangent space is defined via differentiable curves, then this map is defined by If, instead, the tangent space is defined via derivations, then this map is defined by The linear map is called variously the derivative, total derivative, differential, or pushforward of at . It is frequently expressed using a variety of other notations: In a sense, the derivative is the best linear approximation to near . Note that when , then the map coincides with the usual notion of