Ανάσυρση (pullback)


Ανάσυρση (pullback) Προώθηση (pushforward)


Μαθηματική Ανάσυρση


Μαθηματική Ανάσυρση


Μαθηματική Ανάσυρση Μαθηματική Προώθηση


Ανάσυρση (pullback) Προώθηση (pushforward)

- Μία διαδικασία.


Η ονομασία "Ανάσυρση" σχετίζεται ετυμολογικά με την λέξη "σάρωση".


Στο σχήμα (1) we consider two manifolds M and N, (possibly of different dimension), with coordinate systems xμ and yν, respectively.

We imagine that we have a map:

$ \phi \colon M \rightarrow N $

and a function:

$ f\colon N \rightarrow R $.

It is obvious that we can compose the map φ with the functionf to construct a map:

$ ( f \circ \phi) \colon M \rightarrow R $,

which is simply a function on manifold M.

Such a construction is, sufficiently, useful that it gets its own name; we define the pullback of the function f by the map φ, denoted $ \phi_*f $, by

$ \phi_*f = f \circ \phi $

The name makes sense, since we think of $ \phi_*f $ as "pulling back" the function f from manifold N to manifold M.


In mathematics, a pullback is either of two different, but related processes: precomposition and fibre-product.

Its "dual" is pushforward measure.


Precomposition with a function probably provides the most elementary notion of pullback:

in simple terms, a function f of a variable y, where y itself is a function of another variable x, may be written as a function of x.

This is the pullback of f by the function y.

$ f(y(x)) \equiv g(x) \, $

It is such a fundamental process, that it is often passed over without mention, for instance in elementary calculus: this is sometimes called omitting pullbacks, and pervades areas as diverse as fluid mechanics and differential geometry.

However, it is not just functions that can be "pulled back" in this sense.

Pullbacks can be applied to many other objects such as differential forms and their cohomology classes.



The notion of pullback as a fibre-product ultimately leads to the very general idea of a categorical pullback, but it has important special cases:

The pullback bundle is perhaps the simplest example that bridges

  • the notion of a pullback as precomposition, and
  • the notion of a pullback as a cartesian square.

In that example, the base space of a fiber bundle is pulled back, in the sense of precomposition, above.

The fibers then travel along with the points in the base space at which they are anchored: the resulting new pullback bundle looks locally like a cartesian product of the new base space, and the (unchanged) fiber.

The pullback bundle then has two projections:

  • one to the base space,
  • the other to the fiber;

the product of the two becomes coherent when treated as a fiber product.


Functional analysisEdit

When the pullback is studied as an operator acting on function spaces, it becomes a linear operator, and is known as the composition operator. Its adjoint is the push-forward, or, in the context of functional analysis, the transfer operator.


The relation between the two notions of pullback can perhaps best be illustrated by sections of fibre bundles: if s is a section of a fibre bundle E over N, and f is a map from M to N, then the pullback (precomposition) $ f^* s=s\circ f $ of s with f is a section of the pullback (fibre-product) bundle f*E over M.

pushforward & pullbackEdit

The tangent linear application (often called pushforward) can be defined for a differentiable mapping between manifolds.

Given a vector ($ \scriptstyle \mathbf{v} $) of the tangent space at a point P, an mapping is defined on the set of functions defined in the environment of that point, which assigns to each function (at real values) the directional derivative of the function according to vector $ \scriptstyle \mathbf{v} $

$ \mathbf{v}:C^{(k)}(\mathcal{M})\to \R, \qquad f\mapsto \mathbf{v}(f) $

Taking into account the previous operation of vectors on functions and given the differentiable mapping $ \scriptstyle \phi:\mathcal{M}\to \mathcal{N} $ the tangent linear mapping is defined:

$ \phi_*:T\mathcal{M} \to T\mathcal{N} $

Such that a vector in p $ \scriptstyle (p,\mathbf{v}) $ is assigned the only vector $ \scriptstyle (\phi(p),\mathbf{w}) $ that makes it true that:

$ \mathbf{v}(\tilde{f})|_p = \mathbf{w}(f)|_{\phi(p)},\quad \forall f:\mathcal{N}\to \R $


$ \tilde{f} :=f \circ \phi: \mathcal{M}\to \R \quad, \qquad \phi_*(\mathbf{v})= \mathbf{w} $

Once the tangent linear mapping has been defined, the cotangent (or pullback) mapping can be defined on 1-forms as:

$ \phi^*:T^*\mathcal{N} \to T^*\mathcal{M}, \qquad \forall\omega \in T^*\mathcal{N}: \forall\mathbf{v} \in T\mathcal{M}: \phi^*(\omega)(\mathbf{v}) = (\omega)(\phi_*\mathbf{v}) $


Εσωτερική ΑρθρογραφίαEdit



Ikl Κίνδυνοι ΧρήσηςIkl

Αν και θα βρείτε εξακριβωμένες πληροφορίες
σε αυτήν την εγκυκλοπαίδεια
ωστόσο, παρακαλούμε να λάβετε σοβαρά υπ' όψη ότι
η "Sciencepedia" δεν μπορεί να εγγυηθεί, από καμιά άποψη,
την εγκυρότητα των πληροφοριών που περιλαμβάνει.

"Οι πληροφορίες αυτές μπορεί πρόσφατα
να έχουν αλλοιωθεί, βανδαλισθεί ή μεταβληθεί από κάποιο άτομο,
η άποψη του οποίου δεν συνάδει με το "επίπεδο γνώσης"
του ιδιαίτερου γνωστικού τομέα που σας ενδιαφέρει."

Πρέπει να λάβετε υπ' όψη ότι
όλα τα άρθρα μπορεί να είναι ακριβή, γενικώς,
και για μακρά χρονική περίοδο,
αλλά να υποστούν κάποιο βανδαλισμό ή ακατάλληλη επεξεργασία,
ελάχιστο χρονικό διάστημα, πριν τα δείτε.

Οι διάφοροι "Εξωτερικοί Σύνδεσμοι (Links)"
(όχι μόνον, της Sciencepedia
αλλά και κάθε διαδικτυακού ιστότοπου (ή αλλιώς site)),
αν και άκρως απαραίτητοι,
είναι αδύνατον να ελεγχθούν
(λόγω της ρευστής φύσης του Web),
και επομένως είναι ενδεχόμενο να οδηγήσουν
σε παραπλανητικό, κακόβουλο ή άσεμνο περιεχόμενο.
Ο αναγνώστης πρέπει να είναι
εξαιρετικά προσεκτικός όταν τους χρησιμοποιεί.

- Μην κάνετε χρήση του περιεχομένου της παρούσας εγκυκλοπαίδειας
αν διαφωνείτε με όσα αναγράφονται σε αυτήν


>>Διαμαρτυρία προς την wikia<<

- Όχι, στις διαφημίσεις που περιέχουν απαράδεκτο περιεχόμενο (άσεμνες εικόνες, ροζ αγγελίες κλπ.)

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