State a problem that you consider interesting and that you have not explored in the first test, in the assignments or in the previous items of this assignment.
To give you an idea, these are some of the problems that have already been given:
1 - Create an animation illustrating the behavior of the gradient vector of a function, f, that needs to be optimized within a region defined by a constraint of the form g = const; Illustrate (without animation) the behavior of the gradient vector of a function, f, that needs to be optimized within a region defined by two constraints of the form g = 0 and h = 0;
2 - Select a parametrized curve in space over the interval [0, 1] (excluding a straight line or a circle) and create an animation showing a particle moving along the curve. At each instant, represent its velocity vector and acceleration vector. Determine the length of the curve.
3 - Choose an object composed of at least three distinct surfaces: a bottom, a top, and a lateral surface.
i) Parameterize each of the surfaces with a domain of [0, 1]² and plot them.
ii) Calculate the area of each surface (sketch the integration region on their respective coordinate axes).
iii) Calculate the volume of the object (sketch the integration region).
4 - Consider the vector field \( \mathbf{F}(x, y, z) = \frac{(x, y, z)}{||(x, y, z)||^3} \) and the surface of the ellipsoid \( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \), where \( a, b, c \) are distinct digits of your student ID. Considering the outward orientation, calculate:
\begin{enumerate}
\item the flux through this surface without using Gauss' theorem;
\item the flux through this surface using Gauss' theorem.
\end{enumerate}
Sketch the surface and the vector field.