OEF gradient --- Introduction ---

This module actually contains 4 exercises on gradient of two variables functions.

Courbes paramétrées et gradient

Soit une courbe paramétrée d'équations pour . Soit une fonction de dans telle que pour .

On se donne les valeurs suivantes :

c1()= c2()=
c1()= c2()=
c1'()= c2'()=
c1'()= c2'()=

Que vaut ? Le gradient de au point peut-il être non nul ? Donner les valeurs possibles de la pente du gradient de au point dans le cas où ce gradient est non nul. S'il y a plusieurs possibilités, les écrire toutes (en les séparant par des virgules). En effet, le gradient de au point est nul. On suppose que est de classe . Calculer

et dire si admet un extremum local en

Gradient I

Here are some equidistant level curves of the function defined by . Compute the slope at the point A=(,) at the level curve passing through point A (give out the slope to close, if it is finite and inf if it is infinite).
xrange -, + yrange -, + parallel -,-,+,-,0,/10,20,grey parallel -,-,-,+,/10,0,20,grey arrow 0,0, 0,,10,black arrow 0,0, , 0 ,10,black vline 0,0, black hline 0,0, black levelcurve magenta, , levelcurve blue, , disk ,, 5,blue text black, ,, giant, A

Gradient II

Here are some level curves of defined with drawn with a step and two points and . Is the gradient of of norm at point or at point ?
xrange -, yrange -, parallel -,-,,-, 0,0.5, *20, grey parallel -,-,-,, 0.5,0, *20, grey arrow -,0,,0,10,black arrow 0,-,0,,10,black levelcurve magenta,, disk ,, 5, blue disk ,, 5, blue text black, ,medium, text black, ,medium,

Slope and gradient

You are on the hill of equation at the point of coordinates (,) on the map. In what direction (on the map) are you going if you wish to reach the summit as soon as possible ? Give out your answer as a unitary vector:
( , )
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