Suppose that is a smooth map between smooth manifolds; then the differential of at a point is, in some sense, the best linear approximation of near . It can be viewed as a generalization of the total derivative of ordinary calculus. Explicitly, it is a linear map from the tangent space of at to the tangent space of at . Hence it can be used to push forward tangent vectors on to tangent vectors on .
The differential of a map is also called, by various authors, the derivative or total derivative of , and is sometimes itself called the pushforward.
If a map, , carries every point on manifold to manifold then the pushforward of carries vectors in the tangent space at every point in to a tangent space at every point in .
Motivation. Let be a smooth map from an open subset of to an open subset of . For any point in , the Jacobian of at (with respect to the standard coordinates) is the matrix representation of the total derivative of at , which is a linear map
from to . We wish to generalize this to the case that is a smooth function between any smooth manifolds and .
The differential of a smooth map. Let be a smooth map of smooth manifolds. Given some , the differential (or (total) derivative) of at is a linear map
from the tangent space of at to the tangent space of at . The application of to a tangent vector is sometimes called the pushforward of by . 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 then the differential is given by
Here is a curve in with . In other words, the pushforward of the tangent vector to the curve at is just the tangent vector to the curve at .
Alternatively, if tangent vectors are defined as derivations acting on smooth real-valued functions, then the differential is given by
Here , therefore is a derivation defined on and is a smooth real-valued function on . By definition, the pushforward of at a given in is in and therefore itself is a derivation.
After choosing charts around and , is locally determined by a smooth map
between open sets of and , and has representation (at )
in the Einstein summation notation, where the partial derivatives are evaluated at the point in corresponding to 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 at each point. Therefore, in some chosen local coordinates, it is represented by the Jacobian of the corresponding smooth map from to . In general the differential need not be invertible. If is a local diffeomorphism, then the pushforward at is invertible and its inverse gives the pullback of .
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.