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mathematical physics > differential operator > gradient
mathematical analysis > functional analysis > operator > differential operator > gradient
mathematical physics > vector calculus > vector calculus identities > gradient
mathematical analysis > calculus > vector calculus > vector calculus identities > gradient
geometry > differential geometry > vector calculus > vector calculus identities > gradient

Terme préférentiel

gradient  

Définition(s)

  • In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) whose value at a point is the "direction and rate of fastest increase". The gradient transforms like a vector under change of basis of the space of variables of . If the gradient of a function is non-zero at a point , the direction of the gradient is the direction in which the function increases most quickly from , 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 may be defined by:
    where is the total infinitesimal change in for an infinitesimal displacement , and is seen to be maximal when is in the direction of the gradient . The nabla symbol , written as an upside-down triangle and pronounced "del", denotes the vector differential operator.
    (Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Gradient)

Concept(s) générique(s)

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RDF/XML TURTLE JSON-LD Dernière modif. 12/10/2023