Pushforward (differential)
Suppose that φ : M → N is a smooth map between smooth manifolds; then the differential of φ at a point x is, in some sense, the best linear approximation of φ near x. 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 M at x to the tangent space of N at φ(x). Hence it can be used to push tangent vectors on M forward to tangent vectors on N.
The differential of a map φ is also called, by various authors, the derivative or total derivative of φ, and is sometimes itself called the pushforward.
Let φ : U → V be a smooth map from an open subset U of Rm to an open subset V of Rn. For any point x in U, the Jacobian of φ at x (with respect to the standard coordinates) is the matrix representation of the total derivative of φ at x, which is a linear map
We wish to generalize this to the case that φ is a smooth function between any smooth manifolds M and N.
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 dφx 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).
...
Wikipedia