@prefix psr: . @prefix skos: . psr: a skos:ConceptScheme . psr:-HGBTZV5W-5 skos:definition """Pour une fonction

{\\\\displaystyle f:U\\\\subset \\\\mathbb {R} \\ o \\\\mathbb {R} }

, le gradient de

f

se confond avec la dérivée de

f

. Pour une fonction

{\\\\displaystyle f:U\\\\subset \\\\mathbb {R} ^{n}\\ o \\\\mathbb {R} }

, où

n

est un nombre entier

\geq 2

, le gradient de

f

en un point est un vecteur dont la direction est celle de la variation la plus forte de

f

au voisinage de ce point. Cette notion est liée à celle de différentielle pour des fonctions à valeurs réelles : si

f

est différentiable en

a

, la différentielle

D f (a)

est une forme linéaire ; à cette forme linéaire, si l'ensemble de départ

E

est de dimension finie, on peut associer un vecteur qui est le gradient de

f

a

.
(Wikipedia, L'Encylopédie Libre, https://fr.wikipedia.org/wiki/Gradient)"""@fr, """In vector calculus, the gradient of a scalar-valued differentiable function

{\\\\displaystyle f}

of several variables is the vector field (or vector-valued function)

{\\\\displaystyle \\ abla f}

whose value at a point

{\\\\displaystyle p}

is the "direction and rate of fastest increase". The gradient transforms like a vector under change of basis of the space of variables of

{\\\\displaystyle f}

. If the gradient of a function is non-zero at a point

{\\\\displaystyle p}

, the direction of the gradient is the direction in which the function increases most quickly from

{\\\\displaystyle p}

, and the magnitude of the gradient is the rate of increase in that direction, the greatest absolute directional derivative. Further, a point where the gradient is the zero vector is known as a stationary point. The gradient thus plays a fundamental role in optimization theory, where it is used to maximize a function by gradient ascent. In coordinate-free terms, the gradient of a function

{\\\\displaystyle f(\\\\mathbf {r} )}

may be defined by:

{\\\\displaystyle df=\\ abla f\\\\cdot d\\\\mathbf {r} }

where

{\\\\displaystyle df}

is the total infinitesimal change in

{\\\\displaystyle f}

for an infinitesimal displacement

{\\\\displaystyle d\\\\mathbf {r} }

, and is seen to be maximal when

{\\\\displaystyle d\\\\mathbf {r} }

is in the direction of the gradient

{\\\\displaystyle \\ abla f}

. The nabla symbol

{\\\\displaystyle \\ abla }

, written as an upside-down triangle and pronounced "del", denotes the vector differential operator.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Gradient)"""@en ; skos:broader psr:-RRBN6FVB-9, psr:-P6BGQTWK-G ; a skos:Concept ; skos:prefLabel "gradient"@fr, "gradient"@en ; skos:exactMatch , ; skos:inScheme psr: . psr:-RRBN6FVB-9 skos:prefLabel "opérateur différentiel"@fr, "differential operator"@en ; a skos:Concept ; skos:narrower psr:-HGBTZV5W-5 . psr:-P6BGQTWK-G skos:prefLabel "vector calculus identities"@en, "identités vectorielles"@fr ; a skos:Concept ; skos:narrower psr:-HGBTZV5W-5 .