to , thus turning the latter set into an -dimensional real vector space. Again, one needs to check that this construction does not depend on the particular chart and the curve being used, and in fact it does not. Definition via derivations Suppose now that is a manifold. A real-valued function is said to belong to if and only if for every coordinate chart , the map is infinitely differentiable. Note that is a real associative algebra with respect to the pointwise product and sum of functions and scalar multiplication. A derivation at is defined as a linear map that satisfies the Leibniz identity which is modeled on the product rule of calculus. (For every identically constant function it follows that ). Denote the set of all derivations at Setting and turns into a vector space. Generalizations Generalizations of this definition are possible, for instance, to complex manifolds and algebraic varieties. However, instead of examining derivations from the full algebra of functions, one must instead work at the level of germs of functions. The reason for this is that the structure sheaf may not be fine for such structures. For example, let be an algebraic variety with structure sheaf . Then the Zariski tangent space at a point is the collection of all -derivations , where is the ground field and is the stalk of at . Equivalence of the definitions For and a differentiable curve such that define (where the derivative is taken in the ordinary sense because is a function from to ). One can ascertain that is a derivation at the point and that equivalent curves yield the same derivation. Thus, for an equivalence class we can define where the curve has been chosen arbitrarily. The map is a vector space isomorphism between the space of the equivalence classes and that of the derivations at the point Definition via cotangent spaces Again, we start with a manifold and a point . Consider the ideal of that consists of all smooth functions vanishing at , i.e., . Then and are both real vector spaces, and the quotient space can be shown to be isomorphic to the cotangent space through the use of Taylor's theorem. The tangent space may then be defined as the dual space of . While this definition is the most abstract, it is also the one that is most easily transferable to other settings, for