OEF gradient --- Introduction ---

This module actually contains 5 exercises on the gradient of 2 variables fonctions.

Parametric curve and gradient

Let [,] to be a parametric curve of equations

for in [,].

and let be a function from to such that

for in [,].

We give the following values :

What is the value of ? Can the gradient of at point be non zero ? Give the possible values of the slope of the gradient of at point in the case when it is non zero. If there is several cases, write all values (separate them with comma). Indeed the gradient of at point is zero. We assume that is in . Compute

and decide if admits a local extremum at

Gradient I

Here are some equidistant level curves of the function defined by


Compute the direction at the point at the level curve passing through point (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 et are given and drawn. Is the gradient of of larger 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,

Isotherms and adiabatics

We call isotherms the level curves of the function


and adiabatics the level curves of the function

to .

The level curves of and are drawn. the are .

xrange 0,2* yrange -/10,2* levelcurve ,y*x^(), levelcurve ,y*x^(), hline black, 0,0 vline black, 0,0 text black, *1.1,*1.1,medium, A disk ,,7,blue linewidth 2 line -1,-, +1,+, line -1,-, +1,+,
The exercise has several steps.

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 vector:

( , )

What is the angle of your starting direction on the hill and ? Give the answer in degrees and approximate it with the nearest real number with one decimal

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Description: collection of exercises on the gradient of 2 variables functions. This is the main site of WIMS (WWW Interactive Multipurpose Server): interactive exercises, online calculators and plotters, mathematical recreation and games

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