Pushforward (differential) - The Differential of A Smooth Map

The Differential of A Smooth Map

Let φ : M → N be a smooth map of smooth manifolds. Given some x ∈ M, the differential of φ at x is a linear map

from the tangent space of M at x to the tangent space of N at φ(x). The application of dφ_x to a tangent vector X is sometimes called the pushforward of X by φ. The exact definition of this pushforward depends on the definition one uses for tangent vectors (for the various definitions see tangent space).

If one defines tangent vectors as equivalence classes of curves through x then the differential is given by

Here γ is a curve in M with γ(0) = x. In other words, the pushforward of the tangent vector to the curve γ at 0 is just the tangent vector to the curve φ∘γ at 0.

Alternatively, if tangent vectors are defined as derivations acting on smooth real-valued functions, then the differential is given by

Here X ∈ T_xM, therefore X is a derivation defined on M and f is a smooth real-valued function on N. By definition, the pushforward of X at a given x in M is in T_φ(x)N and therefore itself is a derivation.

After choosing charts around x and φ(x), F is locally determined by a smooth map

between open sets of Rm and Rn, and dφ_x has representation (at x)

in the Einstein summation notation, where the partial derivatives are evaluated at the point in U corresponding to x in the given chart.

Extending by linearity gives the following matrix

Thus the differential is a linear transformation, between tangent spaces, associated to the smooth map φ at each point. Therefore, in some chosen local coordinates, it is represented by the Jacobian of the corresponding smooth map from Rm to Rn. In general the differential need not be invertible. If φ is a local diffeomorphism, then the pushforward at x is invertible and its inverse gives the pullback of T_φ(x)N.

The differential is frequently expressed using a variety of other notations such as

It follows from the definition that the differential of a composite is the composite of the differentials (i.e., functorial behaviour). This is the chain rule for smooth maps.

Also, the differential of a local diffeomorphism is a linear isomorphism of tangent spaces.