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A tangent vector at a point p on a smooth manifold is a derivation — a linear map from smooth functions to real numbers satisfying the Leibniz (product) rule. The tangent space TpM is the vector space of all tangent vectors at p. This definition avoids embedding the manifold in an ambient space, making it intrinsic. In local coordinates, tangent vectors correspond to directional derivatives ∂/∂xⁱ, and TpM has the same dimension as M.
In multivariable calculus, the tangent space at a point on a surface in ℝ³ is a plane that just touches the surface — a linear approximation to the surface near that point. But this picture depends on the surface sitting inside ℝ³. On an abstract smooth manifold, there is no ambient space to host a tangent plane. The derivation definition solves this by defining tangent vectors in terms of what they *do* rather than where they *live*: a tangent vector acts on smooth functions by differentiating them.
Formally, a derivation at p ∈ M is a linear map v : C∞(M) → ℝ satisfying the Leibniz rule v(fg) = f(p)v(g) + g(p)v(f). The set of all derivations at p forms a vector space under pointwise addition and scalar multiplication — this is the tangent space TpM. Two immediate consequences of the Leibniz rule: derivations kill constants (v(c) = 0 for any constant function c), and derivations depend only on the local behavior of f near p (if f = g on a neighborhood of p, then v(f) = v(g)).
In a coordinate chart (U, φ) with coordinates (x¹, ..., xⁿ), the operators ∂/∂xⁱ|_p are derivations. The operator ∂/∂xⁱ|_p acts on f by computing the iᵗʰ partial derivative of f ∘ φ⁻¹ at φ(p). These n derivations form a basis for TpM, so dim(TpM) = dim(M). A general tangent vector is v = vⁱ ∂/∂xⁱ|_p, where the coefficients vⁱ are the components of v in this coordinate basis. Under a change of coordinates (x¹, ..., xⁿ) → (y¹, ..., yⁿ), the components transform by the Jacobian matrix: v's components in the y-basis are (∂yʲ/∂xⁱ)vⁱ. This transformation law is the classical definition of a "contravariant vector."
A smooth map F : M → N between manifolds induces a linear map dFp : TpM → TF(p)N called the differential (or pushforward). It acts by (dFp(v))(g) = v(g ∘ F) for any smooth function g on N. In coordinates, dFp is represented by the Jacobian matrix of F. The differential is the manifold version of the total derivative from multivariable calculus, and it is the primary tool for relating the geometry of different manifolds. If F is a diffeomorphism, then dFp is an isomorphism of tangent spaces.
The tangent bundle TM is the disjoint union of all tangent spaces: TM = ∪_p TpM. It is itself a smooth manifold of dimension 2n (n coordinates for the base point, n for the tangent vector). A smooth assignment of a tangent vector to each point of M — a section of the tangent bundle — is a vector field, which is the next major concept in differential geometry. The tangent space construction is the gateway to all the linear algebra that happens "fiberwise" on a manifold.