Manifolds 23 | Differential (Definition) [dark version]

Differentials for smooth maps between manifolds are built by pushing tangent vectors forward—turning the familiar “Jacobian/derivative” idea into a coordinate-free construction. For a smooth map F: M → N, each point p ∈ M has a tangent space T_pM, and the map induces a linear map dF_p: T_pM → T_{F(p)}N. The key move is that tangent vectors are represented as equivalence classes of curves through p, so dF_p is defined by composing those curves with F and taking the resulting tangent class on the target manifold. This makes the differential the manifold version of a linear approximation of F near p, just like the derivative does in ordinary calculus.

To organize all tangent spaces at once, the transcript introduces the tangent bundle TM as the disjoint union of tangent spaces over every point: TM = ⨆_{p∈M} T_pM. Visually, each tangent space “sits over” its base point, and the disjoint union keeps tangent spaces at different points from being accidentally identified. The tangent bundle can itself be given a smooth manifold structure, with dimension equal to twice the dimension of M—consistent with the idea that it behaves like a product locally (one set of coordinates for the base point, another for the tangent directions).

After setting up tangent spaces and the tangent bundle, the differential is defined precisely using the curve-based description of tangent vectors. A tangent vector at p is an equivalence class of curves γ with γ(0)=p. The differential sends that class to the equivalence class of the composed curve F∘γ, producing a well-defined linear map between vector spaces. Because this construction depends on p, the notation dF_p keeps the base point explicit.

The transcript then connects the abstract definition to the concrete multivariable calculus formula. When M and N are smooth embedded submanifolds of R^n, the abstract tangent spaces can be identified with the usual tangent spaces in Euclidean space. In that setting, the differential becomes the familiar Jacobian action: for a C^1 map f: R^n → R, applying d f_p to a tangent vector represented by γ corresponds to the directional derivative of f along that vector. Using the chain rule, the resulting expression is the Jacobian matrix of f at p multiplied by the derivative of γ at 0, which is exactly the standard multivariable calculus computation.

Overall, the differential on manifolds is presented as the correct generalization of the Jacobian matrix and the total derivative: it produces directional derivatives in a coordinate-free way, while still reducing to the Jacobian in local Euclidean coordinates. The next step, hinted at for later videos, is to examine how this differential looks in local charts, where it should mirror the Jacobian-matrix behavior even in manifold language.