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Concept
Pushforward (differential)
Summary
In differential geometry, pushforward is a linear approximation of smooth maps on tangent spaces. Suppose that is a smooth map between smooth manifolds; then the differential of at a point , denoted , 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, the differential is a linear map from the tangent space of at to the tangent space of at , . Hence it can be used to push tangent vectors on forward to tangent vectors on . The differential of a map is also called, by various authors, the derivative or total derivative of .
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
between their tangent spaces. Note the tangent spaces are isomorphic to and , respectively. The pushforward generalizes this construction to the case that is a smooth function between any smooth manifolds and .
Let be a smooth map of smooth manifolds. Given the differential of at is a linear map
from the tangent space of at to the tangent space of at The image of a tangent vector under 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 tangent vectors are defined as equivalence classes of the curves for which then the differential is given by
Here, is a curve in with and is tangent vector to the curve at In other words, the pushforward of the tangent vector to the curve at is 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
for an arbitrary function and an arbitrary derivation at point (a derivation is defined as a linear map that satisfies the Leibniz rule, see: definition of tangent space via derivations).
Official source
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