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Nova strana: {{Prijevod}} {{Drugo_značenje|naslov=Gradijent|Gradijent}} [[Slika:Gradient2.svg|thumb|300px|Na gornjim slikama, skalarno polje prikazano je crnom i bijelom područijem, s tim da crna ...
 
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Gradijent (ili gradijent vektorskog polja) skalarne funkcije <math>f(x)</math> po vaktorskoj varijabli <math>x = (x_1,\dots,x_n)</math> se označava kao <math>\nabla f</math> ili <math>\vec{\nabla} f</math> gdje je <math>\nabla</math> ([[nabla simbol]]) označava vektorski [[diferencijalni operator]], [[nabla operator]]. Oznaka <math>\operatorname{grad}(f)</math> se, također, koristi za označavanje gradijenta.
 
ByPrema definitiondefiniciji, thegradijent gradient is aje [[vectorvektorsko fieldpolje]] whose componentsčije aresu thekomponente [[partial derivative|partial derivatives ]] offunkcije <math>f</math>. ThatTo isjest:
: <math> \nabla f = \left(\frac{\partial f}{\partial x_1 }, \dots, \frac{\partial f}{\partial x_n } \right). </math>
(Here the gradient is written as a row vector, but it is often taken to be a column vector; note also that when a function has a time component, the gradient often refers simply to the vector of its spatial derivatives only.)
 
[[Skalarni proizvod]] <math>(\nabla f)_x\cdot v</math> gradijenta u tački ''x'' sa vektorom ''v'' daje [[izvod po pravcu]] funkcije ''f'' u ''x'' u pravcu ''v''.
The [[dot product]] <math>(\nabla f)_x\cdot v</math> of the gradient at a point ''x'' with a vector ''v'' gives the [[directional derivative]] of ''f'' at ''x'' in the direction ''v''. It follows that the gradient of ''f'' is [[orthogonal]] to the [[level set]]s of ''f''. This also shows that, although the gradient was defined in terms of coordinates, it is actually invariant under [[Orthogonal matrix|orthogonal transformation]]s, as it should be, in view of the geometric interpretation given above.
 
Gradijent je [[nerotaciono vektorsko polje]], te su linijski intergrali kroz gradientno polje nezavisni i mogu se izračunati pomoći [[gradijentna teorema|gradijentnom teoremom]]. Suprotno, nerotacijsko vektorsko polje u [[jednostavno poezani prostor|jednostvno povezanom regionu]] je uvijek gradijent funkcije.
Because the gradient is orthogonal to level sets, it can be used to construct a vector normal to a surface. Consider any manifold that is one dimension less than the space it is in (i.e., a surface in 3D, a curve in 2D, etc.). Let this manifold be defined by an equation e.g. ''F''(''x'', ''y'', ''z'') = 0 (i.e., move everything to one side of the equation). We have now turned the manifold into a level set. To find a normal vector, we simply need to find the gradient of the function ''F'' at the desired point.
 
The gradient is an [[irrotational vector field]] and line integrals through a gradient field are path independent and can be evaluated with the [[gradient theorem]]. Conversely, an irrotational vector field in a [[simply connected]] region is always the gradient of a function.
 
== Izrazi za gradijent u 3 dimenzije==