#
OEF gradient
--- Introduction ---

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

### Parametric curve and gradient

Let
[,]
be a parametric curve
of equations
for
[,]. and let
be a function
from
to
such that
for
[,]. 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
.
- On what is the point
? (give the level).
Compute the slope
of this curve en
?
- On what is the point
? (give the level).
Compute the slope
of this curve en
?
- What is the value of
?

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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