- Ένας Τελεστής.
Υπάρχουν πολλά είδη διαφορικών τελεστών
- η Χρονική Παράγωγος, η Χωρική Παράγωγος
- η Ολική Παράγωγος, η Μερική Παράγωγος
- η Κλίση (grad)
- η Απόκλιση (div)
- η Στροβιλισμός (curl)
- η Περιστροφή (rot)
It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function (in the style of a higher-order function in computer science).
This article considers mainly linear operators, which are the most common type. However, non-linear differential operators, such as the Schwarzian derivative also exist.
Assume that there is a map from a function space to another function space and a function so that is the image of i.e.,
A differential operator is represented as a linear combination, finitely generated by and its derivatives containing higher degree such as
where the set of non-negative integers, , is called a multi-index, called length, are functions on some open domain in n-dimensional space and The derivative above is one as functions or, sometimes, distributions or hyperfunctions and or sometimes, .
The most common differential operator is the action of taking the derivative itself.
Common notations for taking the first derivative with respect to a variable x include:
When taking higher, nth order derivatives, the operator may also be written:
The derivative of a function f of an argument x is sometimes given as either of the following:
The D notation's use and creation is credited to Oliver Heaviside, who considered differential operators of the form
in his study of differential equations.
One of the most frequently seen differential operators is the Laplacian operator, defined by
Another differential operator is the Θ operator, or theta operator, defined by
This is sometimes also called the homogeneity operator, because its eigenfunctions are the monomials in z:
In n variables the homogeneity operator is given by
In writing, following common mathematical convention, the argument of a differential operator is usually placed on the right side of the operator itself.
Sometimes an alternative notation is used: The result of applying the operator to the function on the left side of the operator and on the right side of the operator, and the difference obtained when applying the differential operator to the functions on both sides, are denoted by arrows as follows:
Such a bidirectional-arrow notation is frequently used for describing the probability current of quantum mechanics.
Αν και θα βρείτε εξακριβωμένες πληροφορίες
"Οι πληροφορίες αυτές μπορεί πρόσφατα
Πρέπει να λάβετε υπ' όψη ότι
- Μην κάνετε χρήση του περιεχομένου της παρούσας εγκυκλοπαίδειας
αν διαφωνείτε με όσα αναγράφονται σε αυτήν
- Όχι, στις διαφημίσεις που περιέχουν απαράδεκτο περιεχόμενο (άσεμνες εικόνες, ροζ αγγελίες κλπ.)