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The Tangent Bundle, Vector Fields & the Differential

Once a manifold is smooth, every point grows its own tangent space — glue them together and you get the tangent bundle, the stage on which vector fields live and the differential of a map acts. This guide makes those three objects concrete and shows how they fit.

From one tangent space to a whole bundle

In the previous guide we fixed what a smooth manifold M really is: a space covered by charts whose transition maps are smooth. At each point p in M sits a vector space, the tangent space T_p M, of the same dimension n as M — think of it as the plane of all velocities of curves passing through p. So far each T_p M is an isolated object. The leap of this guide is to glue every tangent space together into a single smooth manifold of its own.

That glued object is the tangent bundle TM: as a set it is the disjoint union of all the T_p M, but it carries its own smooth structure of dimension 2n. A point of TM is a pair (p, v) — a base point p together with a tangent vector v in T_p M. The map that forgets the vector, pi(p, v) = p, is the projection TM -> M, and the fiber over p, namely pi^(-1)(p), is exactly the n-dimensional vector space T_p M. This is the first and friendliest example of a vector bundle.

Why is TM smooth and not just a heap of planes? Each chart (U, x^1, ..., x^n) on M hands you a basis of coordinate directions d/dx^1, ..., d/dx^n at every point of U, so any tangent vector over U reads as v = v^1 d/dx^1 + ... + v^n d/dx^n. The 2n numbers (x^1, ..., x^n, v^1, ..., v^n) are coordinates on pi^(-1)(U) inside TM. Where two charts overlap, the v^i transform by the Jacobian of the coordinate change — a smooth, invertible, linear map on each fiber — and that is precisely what makes TM a manifold rather than a jumble.

Trivial locally, twisted globally

Locally TM always looks boring: over a single chart U it is just U x R^n, a slab of parallel copies of R^n. This is called a local trivialization, and it is the defining promise of any vector bundle. The interesting question is global: can you find ONE trivialization over all of M at once? When you can, TM is the product M x R^n and we call M parallelizable.

Usually you cannot — and that failure is geometry, not bookkeeping. The circle S^1 is parallelizable (one nowhere-zero tangent vector field circulates around it), and so, remarkably, is the torus and the 3-sphere S^3. But the 2-sphere S^2 is NOT: the hairy ball theorem says every tangent vector field on S^2 must vanish somewhere, so you can never comb a global frame. The bundle TM remembers this obstruction; the product M x R^n forgets it. That is the whole reason we bother with bundles instead of products.

Vector fields: smooth sections of TM

Now that TM exists, a vector field X has a clean definition: it is a smooth section of the bundle, that is, a smooth map X: M -> TM with pi(X(p)) = p for every p. In words, X assigns to each point p a tangent vector X_p in that point's own tangent space T_p M, and the assignment varies smoothly. Picture an arrow planted at every point of M, all turning gently as you move — the wind across a weather map is the honest mental image.

There is a second, equivalent face of a vector field that you will use constantly: X is a derivation on smooth functions. Given a smooth f: M -> R, the field eats f and produces a new function (Xf)(p) = the directional derivative of f along X_p. In a chart this is just X(f) = X^1 df/dx^1 + ... + X^n df/dx^n, and it obeys the Leibniz rule X(fg) = f X(g) + g X(f). "Arrow at every point" and "first-order differential operator" are two descriptions of the very same thing — switch between them freely.

Two vector fields do NOT commute as operators, and the failure has a name: the Lie bracket [X, Y] = XY - YX. A priori XY looks like a second-order operator, but the second derivatives cancel and [X, Y] is again an honest vector field. Geometrically it measures how much you drift if you flow a little along X, then Y, then back along X, then back along Y — the bracket is the leading-order failure of that square to close. We meet flows and the bracket in full in guide 5 of this rung.

The differential: how a map moves tangent vectors

A smooth map f: M -> N should not just move points; it should move velocities too. Its differential at p, written df_p or f_*, is the linear map df_p: T_p M -> T_{f(p)} N that takes the velocity of a curve through p to the velocity of its image curve through f(p). This is the right, coordinate-free generalization of the Jacobian matrix from multivariable calculus — and in coordinates it IS exactly the Jacobian of f.

f: M -> N,   curve c(t) in M with c(0) = p

  df_p( c'(0) )  =  (f o c)'(0)        (push the velocity forward)

in charts:   df_p = Jacobian( y(x) ),   (df)^a_i = d y^a / d x^i

  X vector field on M  =>  f_* X  defined only if f is a diffeomorphism
The differential sends the velocity of c to the velocity of f composed with c; in charts it is the Jacobian.

Bundling all the df_p together gives a single smooth map Tf: TM -> TN between the total spaces, the global differential (also called the pushforward on tangent vectors). Watch the direction carefully: tangent vectors push FORWARD along f, the same way f points. The rank of df_p — how much of T_p M survives — is the single most important local invariant of a map, and the entire next guide on immersions, submersions, and the regular value theorem is the study of that rank.

Putting the three together

Here is the picture in one breath. The tangent bundle TM is the stage; a vector field is a smooth way of choosing one point on each fiber-stage; and the differential df is how a map between manifolds drags those stages and their chosen vectors along. To see them cooperate, walk through how a vector field is born from a flow and then read off by the differential.

  1. Start with a flow: a smooth family of maps phi_t: M -> M with phi_0 = identity, like the wind blowing every point downstream for time t.
  2. Differentiate in time at t = 0: the velocity X_p = d/dt|_(t=0) phi_t(p) lands in T_p M. Doing this at every p builds a vector field X — the generator of the flow.
  3. Now apply a smooth map f: M -> N to the moving points. The chain rule on the curve t -> f(phi_t(p)) is exactly df_p acting on X_p: the differential reads the velocity in N.
  4. If f is a diffeomorphism, these read-off velocities themselves assemble into a genuine vector field f_* X on N — the pushed-forward field. If not, you only get vectors point-by-point, not a field.

One honest caveat before we move on. Everything here is local-to-global glue plus linear algebra on fibers; we have stated the hairy ball theorem and the existence of TM as a smooth manifold without proving them, because each is a short chapter in Lee or Tu, not a slogan. The payoff is enormous: with TM, vector fields, and df in hand, the next guides can finally speak of submanifolds via the regular value theorem and, soon after, of integrating forms over an oriented manifold.