(PLEASE USE MATLAB IN THIS QUESTION)

We will now display a contour plot along with the gradient vector

field. To display a vector field we use the command ``quiver''

and to display a contour line we use the command ``contour''.

Let f(x, y) = xy-x^3/3. Then f_x(x, y) = y-x^2 and f_y = x. We shall

display the gradient

vector field and the level curves of f over the square

[-2, 2]x[-2, 2]. Run the script

f=inline('x.*y-(x.^3)/3','x','y')

fx=inline('y-x.^2','x','y')

fy=inline('x','x','y')

x=-2:.05:2;y=x;

% this is the ne mesh for the level curves.

[X,Y]=meshgrid(x,y);

Z=f(X,Y);

% We choose the level curves.

levels = [-6:.5:6];

contour(X,Y,Z,levels)

hold on

xx=-2:.2:2; yy=xx;

% This is the coarse mesh for the arrows

[XX,YY]=meshgrid(xx,yy);

U=fx(XX,YY); V=fy(XX,YY);

quiver(XX,YY,U,V)

axis equal

What is the relation between the level curves and the arrows?

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6. Repeat problem 5 for f(x, y) = x^2 + 4 y^2.

Use the same square but you will need to

consider a different set of level curves.

I am having difficulty considering a different set of level curves which would satisfy the function f(x). Can someone explain the logic behind choosing a set of level curves to me?