gradient, plan tangent, courbes de niveau (et fonctions implicites)…

à chaque fois ça me puzzle, pourtant c’est évident !
le gradient est normal à la courbe de niveau, tout simplement parce que c’est leur raison d’être : la courbe de niveau indique le chemin sur une surface qui permet de rester à valeur constante, le gradient est le vecteur qui indique la variation (et qui plus est, la direction de plus forte variation). - ça c’est pour la philosophie -

si on fait un développement limité, pour X dans Rn, t un réel et v un vecteur de Rn, on a f(x+tv) = f(x) + t∇f(x).v + reste , sur la courbe de niveau on a justement f(x+tv) = f(x) donc ∇f(x) est orthogonal à v, pour tout v tel que x+tv est sur la courbe de niveau - ça c’est pour l’analyse -

mais ce qui me puzzle vraiment, plus précisément, c’est la relation avec le plan tangent, certainement du fait de mon inculture géométrique, j’en ai toujours été très complexée.

les dérivées partielles correspondent séparément à la pente de la tangente le long de la courbe donnée par l’intersection de la surface définie par l’équation de la fonction et du plan obtenu fixant la variable qu’on ne va pas dériver au point considéré. ex : z = f(x,y) pour x0 et y0, l’intersection de la surface représentant f et du plan y = y0 est la courbe donnée par la fonction qui à x associe f(x,y0), la tangente à cette courbe en x0 sera donnée par :
z = f(x0,y0) + δf⁄δx (x0,y0) (x -x0 )
de même, la tangeante à la courbe de la fonction qui à y associe f(x0,y) en y0 sera donnée par :
z = f(x0,y0) + δf⁄δy (x0,y0) (y -y0 )
le plan déterminé par les deux tangentes est alors donné par :
z = f(x0,y0)+ δ f⁄δx (x0,y0) (x -x0 ) + δ f⁄δ y (x0,y0) (y -y0 )
(ce qui signifie en gros qu’une fonction de R² sera dérivable si ce plan constitue une bonne approximation locale de la fonction, et pas uniquement dans les directions x et y)
de l’équation du plan tangeant, on tire donc son vecteur normal : (δf⁄δx, δf⁄δy, -1) soit ( ∇f(x), -1) .

plantangent_450.jpg

Par ailleurs, intéressons-nous aux courbes de niveau, c’est à dire, pour a donné, l’ensemble des points du plan (x,y) tels que f(x,y) = a. Le théorème des fonctions implicites nous dit justement que si les dérivées partielles de f existent et sont continues dans un voisinage d’un point (x0,y0) de la courbe de niveau a avec la dérivée partielle en y non nulle, on sait qu’il existe des voisinages U de x0 et V de y0 et une unique fonction y(x) de U dans V tels que f(x,y(x))=a. De plus, cette fonction y est différentiable et satisfait y’(x) = - δf(x,y(x))⁄δx / δf(x,y(x))⁄δy. Ceci nous donne le vecteur directeur de la tangente à la ligne de niveau en (x0,y0) : (1, - δf(x,y(x))⁄δx / δf(x,y(x))⁄δy) *. Par ailleurs la projection sur le plan (x,y) du vecteur normal au plan tangent est donné tout simplement par le gradient (δf(x,y(x))⁄δx , δf(x,y(x))⁄δy), on voit bien qu’il est alors orthogonal à la ligne de niveau ! la boucle est bouclée.

(il n’est bien sûr jamais trop tard pour réaliser aussi qu’en 1D, l’équation de la tangente est donné par y = f’(x0)(x-x0) + f(x0), elle a bien pour vecteur directeur (1,f’(x0)) et pour vecteur normal (f’(x0), -1).)
(* oui et donc l’équation de la tangente à la courbe de niveau est bien l’intersection du plan tangent avec le plan (x0y), reporter dans l’équation du plan tangent z=0 )

(référence de tout cela et copyright du stéréogramme - quelle bonne idée ! - , -il y en a d’autres dans le livre-, toujours l’excellentissime “Analyse au fil de l’histoire” de E. Hairer et G. Wanner)

Je me sens mieux.

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