the pushforward of a smooth map

A smooth map f: M -> N takes points to points. The pushforward is its 'linearization': at each point it tells you how f bends infinitesimal arrows, turning a velocity at p into a velocity at f(p). It is the manifold version of the Jacobian matrix, and it is the single device that lets all of calculus migrate from R^n onto manifolds.

At a point p, the pushforward (also called the differential) is a linear map df_p: T_p M -> T_{f(p)} N. The cleanest definition is via curves: if v is the velocity at time 0 of a curve gamma through p, then df_p(v) is the velocity at time 0 of the image curve f composed with gamma. Equivalently, on functions, (df_p(v))(g) = v(g composed with f) for every smooth g on N — the pushed-forward vector differentiates functions on N by first pulling them back through f. In coordinates df_p is exactly the Jacobian matrix of partial derivatives of f's components, so this abstract map reduces to ordinary multivariable calculus once charts are chosen. Assembling all the df_p gives a smooth bundle map df: TM -> TN covering f.

The pushforward is the engine behind almost every theorem in the field. Its rank at p — the dimension of the image of df_p — classifies the local behavior of f: maximal rank equal to dim N makes f a submersion, maximal rank equal to dim M makes it an immersion, and df_p being an isomorphism makes f a local diffeomorphism (inverse function theorem). One caution: the pushforward of a vector field along f generally does NOT define a vector field on N — that only works when f is a diffeomorphism, since otherwise different points of M can map to one point of N with incompatible images, or miss points of N entirely. Covectors, by contrast, always pull back.