Διαφορικό

Διαφορικόν

Differential

Διαφορικό

Διαφορικό
Παράγωγος

Διαφορικό

Διαφορική Τοπολογία
Διαφορική Γεωμετρία

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Διαφορική Μορφή
Μοναδιαίο Διάνυσμα

Διαφορικό

- Μία Μαθηματική έννοια.

Ετυμολογία

Η ονομασία "Διαφορικό" σχετίζεται ετυμολογικά με την λέξη "Διαφορά".

Εισαγωγή

In mathematics, the term differential has several meanings.

Basic notions

In calculus, the differential represents a change in the linearization of a function. In traditional approaches to calculus, the differentials (e.g. dx, dy, dt etc...) are interpreted as infinitesimals.

There are several methods of defining infinitesimals rigorously, but it is sufficient to say that

• an infinitesimally small number is smaller than any real, positive number, and
• an infinitely large one is larger than any real number.

The differential is another name for the Jacobian matrix of partial derivatives of a function from Rⁿ to R^m (especially when this matrix is viewed as a linear map).

More generally,

• the differential or pushforward refers to the derivative of a map between smooth manifolds and the pushforward operations it defines.
• The differential is also used to define the dual concept of pullback.

Stochastic calculus provides a notion of stochastic differential and an associated calculus for stochastic processes.

The integrator in a Stieltjes integral is represented as the differential of a function.

Formally, the differential appearing under the integral behaves exactly as a differential, thus,

• the "integration by substitution" and
• "integration by parts" formulae

for Stieltjes integral

correspond, respectively, to:

• the chain rule and
• the product rule

for the differential.

Differential geometry

The notion of a differential motivates several concepts in differential geometry (and differential topology).

• Differential forms provide a framework which accommodates multiplication and differentiation of differentials.
• The exterior derivative is a notion of differentiation of differential forms which generalizes the differential of a function (which is a differential 1-form).
• Pullback is, in particular, a geometric name for the chain rule for composing a map between manifolds with a differential form on the target manifold.
• Covariant derivatives or differentials provide a general notion for differentiating of vector fields and tensor fields on a manifold, or, more generally, sections of a vector bundle: see Connection . This ultimately leads to the general concept of a connection.

Algebraic geometry

Differentials are also important in algebraic geometry, and there are several important notions.

• Abelian differentials usually refer to differential one-forms on an algebraic curve or Riemann surface.
• Quadratic differentials (which behave like "squares" of abelian differentials) are also important in the theory of Riemann surfaces.
• Kähler differentials provide a general notion of differential in algebraic geometry

Other meanings

The term differential has also been adopted in homological algebra and algebraic topology, because of the role the exterior derivative plays in de Rham cohomology: in a cochain complex $(C_{\bullet },d_{\bullet })$, the maps (or coboundary operators) d_i are often called differentials. Dually, the boundary operators in a chain complex are sometimes called codifferentials.

The properties of the differential also motivate the algebraic notions of a derivation and a differential algebra.

Θεώρηση Spivak

Classical differential geometers (and classical analysts) did not hesitate to talk about
"infinitely small" changes dx, dy, dz of the coordinates x, y, z, just as Leibnitz had.

No one wanted to admit that this was nonsense,
because true results were obtained when these infinitely small quantities were divided into each other
(provided one did it in the right way).

Eventually it was realized that the closest one can come to describing
an infinitely small change is to describe a direction in which
this change is supposed to occur, i.e., a tangent vector.

Since change df is supposed to be the infinitesimal change of function f under an infinitesimal change of the point,
then, change df must be a function of this change,
which means that df should be a function on tangent vectors.

The changes dx, dy, dz, themselves, then metamorphosed into functions, and
it became clear that they must be distinguished from the tangent vectors ∂/∂x, ∂/∂y, ∂/∂z.

Εσωτερική Αρθρογραφία

• Διαφορικός Τελεστής
• Ανάδελτα
• Διαφορικότητα
• Διαφορισιμότητα
• Ολοκληρωτικότητα
• Ολοκληρωσιμότητα

Ιστογραφία

• Ομώνυμο άρθρο στην Βικιπαίδεια
• Ομώνυμο άρθρο στην Livepedia
• What is the intuition behind the definition of the differential of a function
• Διαφορικό, Τζέμος
• Διαφορική Γεωμετρία, Βλάχος

Κίνδυνοι Χρήσης

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. Επίσης, 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

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