Tangent space
At each point p of a smooth n-manifold M, the n-dimensional vector space T_p M of all tangent vectors at p. Tangent vectors transform as ∂x^μ/∂x'^ν under coordinate change — the prototype of contravariant index behaviour. The disjoint union of all tangent spaces is the tangent bundle TM.
Definition
The tangent space T_p M at a point p of a smooth n-dimensional manifold M is the n-dimensional vector space of all tangent vectors at p. Concretely, a tangent vector at p can be defined as the velocity of a smooth curve passing through p, or equivalently as a linear directional-derivative operator acting on smooth functions defined near p. Every tangent space is isomorphic to ℝⁿ, but the isomorphism depends on choice of coordinates; the basis {∂/∂x^μ} provided by a coordinate chart gives one natural frame, but no privileged one. Vectors in T_p M live at p and at p only — one cannot meaningfully add tangent vectors at distinct points without first specifying a connection that lets one transport them.
The transformation law for tangent vectors under a change of coordinates x → x'(x) is V'^μ = (∂x'^μ/∂x^ν) V^ν — the prototype of "contravariant" index behaviour, the upper index. The dual space, the cotangent space T*_p M, contains the covectors (one-forms) that transform with the inverse Jacobian and carry lower indices. The disjoint union of all tangent spaces over all points of M is the tangent bundle TM, a 2n-dimensional manifold in its own right; sections of TM are vector fields, and the entire calculus of differential geometry is built on the relationship between the manifold, its tangent bundle, and the tensor bundles obtained from tangent and cotangent spaces by tensor product.