Fandom

Science Wiki

Διανυσματική Ανάλυση

63.282pages on
this wiki
Add New Page
Talk1 Share

Ad blocker interference detected!


Wikia is a free-to-use site that makes money from advertising. We have a modified experience for viewers using ad blockers

Wikia is not accessible if you’ve made further modifications. Remove the custom ad blocker rule(s) and the page will load as expected.

Διανυσματική Ανάλυσις

Vector Analysis, Διανυσματικός Λογισμός


- Επιστημονικός Κλάδος των Μαθηματικών

ΕτυμολογίαEdit

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

ΕισαγωγήEdit

Basic objectsEdit

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 operationsEdit

Algebraic operations Edit

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: a \bold{v};
vector addition
addition of two vector fields, yielding a vector field: \bold{v}_1 + \bold{v}_2;
dot product
multiplication of two vector fields, yielding a scalar field: \bold{v}_1 \cdot \bold{v}_2;
cross product
multiplication of two vector fields, yielding a vector field: \bold{v}_1 \times \bold{v}_2;

There are also two triple products:

scalar triple product
the dot product of a vector and a cross product of two vectors: \bold{v}_1\cdot\left( \bold{v}_2\times\bold{v}_3 \right);
vector triple product
the cross product of a vector and a cross product of two vectors: \bold{v}_1\times\left( \bold{v}_2\times\bold{v}_3 \right) or \left( \bold{v}_3\times\bold{v}_2\right)\times\bold{v}_1 ;

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

Differential operations Edit

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

Operation Notation Description Domain/Range
Gradient  \operatorname{grad}(f) = \nabla f Measures the rate and direction of change in a scalar field. Maps scalar fields to vector fields.
Curl  \operatorname{curl}(\mathbf{F}) = \nabla \times \mathbf{F} Measures the tendency to rotate about a point in a vector field. Maps vector fields to (pseudo)vector fields.
Divergence  \operatorname{div}(\mathbf{F}) = \nabla \cdot \mathbf{F} Measures the magnitude of a source or sink at a given point in a vector field. Maps vector fields to scalar fields.
Laplacian  \Delta f = \nabla^2 f = \nabla \cdot \nabla f 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.

TheoremsEdit

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  \int_{L[\mathbf p \to \mathbf q] \subset \mathbb R^n} \nabla\varphi\cdot d\mathbf{r} = \varphi\left(\mathbf{q}\right)-\varphi\left(\mathbf{p}\right)   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  \int\!\!\!\!\int_{A\,\subset\mathbb R^2} \left  (\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right)\, d\mathbf{A}=\oint_{\partial A} \left ( L\, dx + M\, dy \right ) 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  \int\!\!\!\!\int_{\Sigma\,\subset\mathbb R^3} \nabla \times \mathbf{F} \cdot d\mathbf{\Sigma} = \oint_{\partial\Sigma} \mathbf{F} \cdot d \mathbf{r} The integral of the curl of a vector field over a surface in \mathbb R^3 equals the line integral of the vector field over the closed curve bounding the surface.
Divergence theorem

\int\!\!\!\!\int\!\!\!\!\int_{V\,\subset\mathbb R^3}\left(\nabla\cdot\mathbf{F}\right)d\mathbf{V}=\oiint\scriptstyle \partial V\mathbf F\;\cdot{d}\mathbf S

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.

Θέματα - ΤομείςEdit

ΥποσημειώσειςEdit

Εσωτερική ΑρθρογραφίαEdit

ΒιβλιογραφίαEdit

ΙστογραφίαEdit


Ikl.jpg Κίνδυνοι ΧρήσηςIkl.jpg

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

IonnKorr-System-00-goog.png



>>Διαμαρτυρία προς την wikia<<

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


Also on Fandom

Random Wiki