I have to create a visualization of a scalar field given by the formular:
$$f(x,y) = x^3 - 3xy^2$$
I have to represent some features of this scalar field.
I plotted the following scalar fieldbut can´t really say something about its local maximum and minimum.
Every input is appreciated. Thanks in advance.
I plotted the Contour lines of my scalar field but don´t really understand what they tell me about it. It seems to be a 2d view of the field from above.
The jacobian matrix is $O$ when $\frac {\partial f}{\partial x}= 3x^2 -3y^2 =0$ and $\frac {\partial f}{\partial y}=-6xy=0$.
$-6xy=0$ when $x=0$ or $y=0$, and $3x^2 -3y^2 =0$ iff $x= \pm y$, so $x=y=0$.
Thus $(0,0)$ is a critical point. The Hessian doesn't give us any further information, but if you notice: let's restrict $f$ to some lines through the origin, $f(t,t)= -2t^3$; so for $t \gt 0$ we have $f \lt 0$; and for $t \lt 0$ we have $f \gt 0$.
Therefore $(0,0)$ is the only critical point and is also a saddle (without defining any compact set).