The tangent space of a differentiable manifold M is a vector space at a point p on the manifold whose elements are the tangent vectors (or velocities) to the curves passing through that point p. The tangent space at this point p is usually denoted TpM.
The tangent space is necessary for a manifold because it offers a way in which tangent vectors at different points on the manifold can be compared (via an affine connection). If the manifold is a hypersurface of , then the tangent space at a point can be thought of as a hyperplane at that point. However, this ambient Euclidean space is unnecessary to the definition of the tangent space.
The tangent space at a point has the same dimension as the manifold, and the union of all the tangent spaces of a manifold is called the tangent bundle, which itself is a manifold of twice the dimension of M.
Although it is tempting to define a tangent space as a "space where tangent vectors live", without a definition of a tangent space there is no definition of a tangent vector. There are various ways in which a tangent space can be defined, the most intuitive of which is in terms of directional derivatives; the space TpM is the space identified with directional derivatives at p.
 Directional derivative
A curve on the manifold is defined as a differentiable map . Let . If one defines to be all the functions that are differentiable at the point p, then one can interpret
to be a linear functional such that
and is a directional derivative of f in the direction of the curve . This operator can be interpreted as a tangent vector. The tangent space is then the set of all directional derivatives of curves at the point p.
 Directional derivatives as a vector space
If this definition is reasonable, then the directional derivatives, must form a vector space of the same dimension as the n-dimensional manifold M. The easiest way to show this is to show that some n directional derivatives form a basis of the vector space, and in order to do so, one needs to introduce a coordinate chart (see differentiable manifold).
Let where , be a coordinate chart, and . The most obvious candidates for basis vectors would be the directional derivatives along the coordinate curves, i.e. the i-th coordinate curve would be
where , the 1 being in the i-th position.
The directional derivative along a coordinate curve can be represented as
which becomes, via the chain rule,
For an arbitrary curve then
which is simply
as f is arbitrary.
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