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Area Mathematics - General concepts and linear algebra / Scalar and vector fields

IEV ref102-05-19

en
gradient
vector grad f associated at each point of a given space region with a scalar f, having a direction normal to the surface on which the scalar field has a constant value, in the sense of increasing value of f, and a magnitude equal to the absolute value of the derivative of f with respect to distance in this normal direction

Note 1 to entry: The gradient expresses the variation of the scalar field quantity from the given point to a point at an infinitesimal distance ds MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiGacsgacaWGZbaaaa@3732@ in the direction of a given unit vector e by the scalar product df = grad fe ds.

Note 2 to entry: In orthonormal Cartesian coordinates, the three coordinates of the gradient are:

f x , f y , f z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaalaaabaGaeyOaIy RaaGPaVlaadAgaaeaacqGHciITcaaMc8UaamiEaaaacaaMe8UaaGjb VlaaysW7caGGSaGaaGjbVlaaysW7caaMe8UaaGjbVpaalaaabaGaey OaIyRaaGPaVlaadAgaaeaacqGHciITcaaMc8UaamyEaaaacaaMe8Ua aGjbVlaaysW7caGGSaGaaGjbVlaaysW7caaMe8UaaGjbVpaalaaaba GaeyOaIyRaaGPaVlaadAgaaeaacqGHciITcaaMc8UaamOEaaaaaaa@6774@ .

Note 3 to entry: The gradient of the scalar field f is denoted grad f or f MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaGWabiab=DGirlaayI W7caWGMbaaaa@3CD4@ .


fr
gradient, m
vecteur grad f associé en chaque point d'un domaine déterminé de l'espace à un scalaire f, dont la direction est normale à la surface sur laquelle le champ scalaire a une valeur constante, dans le sens des valeurs croissantes de f, et dont la norme est égale à la valeur absolue de la dérivée de f par rapport à la distance dans cette direction normale

Note 1 à l'article: Le gradient exprime la variation du champ scalaire entre le point donné et un point situé à une distance infinitésimale ds MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiGacsgacaWGZbaaaa@3732@ dans la direction d'un vecteur unité donné e par le produit scalaire df = grad fe ds.

Note 2 à l'article: En coordonnées cartésiennes orthonormées, les trois coordonnées du gradient sont:

f x , f y , f z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaalaaabaGaeyOaIy RaaGPaVlaadAgaaeaacqGHciITcaaMc8UaamiEaaaacaaMe8UaaGjb VlaaysW7caGGSaGaaGjbVlaaysW7caaMe8UaaGjbVpaalaaabaGaey OaIyRaaGPaVlaadAgaaeaacqGHciITcaaMc8UaamyEaaaacaaMe8Ua aGjbVlaaysW7caGGSaGaaGjbVlaaysW7caaMe8UaaGjbVpaalaaaba GaeyOaIyRaaGPaVlaadAgaaeaacqGHciITcaaMc8UaamOEaaaaaaa@6774@ .

Note 3 à l'article: Le gradient du champ scalaire f est noté grad f ou f MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaGWabiab=DGirlaayI W7caWGMbaaaa@3CD4@ .


de
Gradient, m

es
gradiente

ja
勾配

pl
gradient

pt
gradiente

sr
градијент, м јд

sv
gradient

zh
梯度

Publication date: 2008-08
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