Διανυσματική Ανάλυσις

Vector Analysis, Διανυσματικός Λογισμός

- Ένας Επιστημονικός Κλάδος των Μαθηματικών

## Ετυμολογία

Η ονομασία "Διανυσματική" σχετίζεται ετυμολογικά με την λέξη "διάνυσμα".

## Basic objects

The basic objects in vector calculus are scalar fields (scalar-valued functions) and vector fields (vector-valued functions). These are then combined or transformed under various operations, and integrated. In more advanced treatments, one further distinguishes pseudovector fields and pseudoscalar fields, which are identical to vector fields and scalar fields except that they change sign under an orientation-reversing map: for example, the curl of a vector field is a pseudovector field, and if one reflects a vector field, the curl points in the opposite direction. This distinction is clarified and elaborated in geometric algebra, as described below.

## Vector operations

### Algebraic operations

The basic algebraic (non-differential) operations in vector calculus are referred to as vector algebra, being defined for a vector space and then globally applied to a vector field, and consist of:

scalar multiplication
multiplication of a scalar field and a vector field, yielding a vector field: ;
addition of two vector fields, yielding a vector field: ;
dot product
multiplication of two vector fields, yielding a scalar field: ;
cross product
multiplication of two vector fields, yielding a vector field: ;

There are also two triple products:

scalar triple product
the dot product of a vector and a cross product of two vectors: ;
vector triple product
the cross product of a vector and a cross product of two vectors: or ;

although these are less often used as basic operations, as they can be expressed in terms of the dot and cross products.

### Differential operations

Vector calculus studies various differential operators defined on scalar or vector fields, which are typically expressed in terms of the del operator (), also known as "nabla". The four most important differential operations in vector calculus are:

Operation Notation Description Domain/Range
Gradient Measures the rate and direction of change in a scalar field. Maps scalar fields to vector fields.
Curl Measures the tendency to rotate about a point in a vector field. Maps vector fields to (pseudo)vector fields.
Divergence Measures the magnitude of a source or sink at a given point in a vector field. Maps vector fields to scalar fields.
Laplacian A composition of the divergence and gradient operations. Maps scalar fields to scalar fields.

where the curl and divergence differ because the former uses a cross product and the latter a dot product, and f denotes a scalar field and F denotes a vector field. A quantity called the Jacobian is useful for studying functions when both the domain and range of the function are multivariable, such as a change of variables during integration.

## Theorems

Likewise, there are several important theorems related to these operators which generalize the fundamental theorem of calculus to higher dimensions:

Theorem Statement Description
Gradient theorem The line integral through a gradient (vector) field equals the difference in its scalar field at the endpoints of the curve L.
Green's theorem The integral of the scalar curl of a vector field over some region in the plane equals the line integral of the vector field over the closed curve bounding the region.
Stokes' theorem The integral of the curl of a vector field over a surface in equals the line integral of the vector field over the closed curve bounding the surface.
Divergence theorem

The integral of the divergence of a vector field over some solid equals the integral of the flux through the closed surface bounding the solid.

## Ιστογραφία

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,
(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.)