Calculus 3: Surface parameterization

Find a parameterization for the cone using cylindrical coordinates, specifically with parameters r and theta, where the height of the cone is $z=r$, and the cone is bounded by $z \leq 3$.

Parametrize the cylindrical surface given by the equation $x^2 + (y - 2)^2 = 4$.

Parametrize the sphere given by the equation $x^2 + y^2 + z^2 = 9$ using spherical coordinates.

Given the parametrization x = u, y = v, and $z=\sqrt{u^2 + v^2}$, determine the rectangular equation.

Given the parametrization $x=3u\sin v$, $y=3u\cos v$, and $z=u^2$, determine the rectangular equation.

Using two parameters, parametrize the surface of a torus (doughnut shape) in three dimensions, considering a circle of radius $a$ at a distance $b$ from the center (z-axis), and varying the parameters to map the surface of the torus.

How do you actually go from an s and a t that goes from $0$ to $2\pi$, in both cases, and turn it into a three-dimensional position vector-valued function that would define this surface?

Determine the resulting shape when both T and S vary freely in the function representing a torus.

Consider the surface given by $x = u \cos(v), y = u \sin(v), z = v$. Set up the integral for the surface integral of a function $f$ over this surface.