Pushforward (differential)

As Push Forward a mapping between tangent spaces of smooth manifolds is called, which generalizes the directional derivative defined in Euclidean space.

The dual concept usually called transport back ( pullback ).

Definition

Are and smooth manifolds and is a smooth map, then we define the push forward

Of the point by

For and any smooth function on the manifold N. These tangent vectors are interpreted as directional derivatives ( derivations ), see tangent space.

In this way, an image is defined.

Names and spellings

Importance for tangent vectors of curves

If the tangent vector of a differentiable curve (this is an interval in ) at the point, then the tangent vector of the curve is in the image pixel, ie

Representation in coordinates

Are local coordinates to local coordinates, and on to the pixel so have the vectors and illustrations

Will continue the mapping represented by the functions so true

Push Forward in Euclidean space

If there is the special case, represents nothing more than the total derivative, in which the Euclidean space is identified in a natural way with its tangent space (the distinction between directional derivative and total derivative does not matter here because the function is already assumed to be sufficiently smooth ).

Often, the tangent space of the Euclidean space in the point identified, the tangent bundle is with words. In this case, the push forward the figure.

Properties

For the Push Forward a concatenation of two figures and the chain rule:

Or pointwise