I would like to know if the following settings are correct:
a) Parameterization of Curve:
Given the curve $C = \{(x, y, z) \in \mathbb{R}^3 \,|\, x^2 + y^2 = e^3\}$ I want to find a parameterization for it.
My proposed parameterization is as follows: Let $x = r \cos(\theta)$ and $y = r \sin(\theta)$, where $r$ is a positive parameter and $\theta$ is an angle parameter. Then:
$x^2 + y^2 = r^2(\cos^2(\theta) + \sin^2(\theta)) = r^2 = e^3$
So, $r = e^{3/2}$ is a constant. The parameterization of $C$ becomes:
$C(t) = (e^{3/2} \cos(t), e^{3/2} \sin(t), z)$
Here, $t$ is the parameter, and $z$ can be any real number since it's not restricted by the equation $x^2 + y^2 = e^3$.
b) Parameterization of Surface:
For the surface $S = \{(x, y, z) \in \mathbb{R}^3 \,|\, x^2 + y^2 = e^z\}$, I'd like to establish a parameterization.
I propose the following parameterization: $S(u, v) = (e^{u/2} \cos(v), e^{u/2} \sin(v), u)$
Here, $u$ and $v$ are the parameters, and $u$ can be any real number since it's not restricted by the equation $x^2 + y^2 = e^z$.
c) Tangent Plane to $S$ at $(0, 1, 0)$:
To determine the tangent plane to the surface $S$ at the point $(0, 1, 0)$, we need to calculate the gradient of the function $F(x, y, z) = x^2 + y^2 - e^z$ and then evaluate it at $(0, 1, 0)$.
The gradient is: $\nabla F(x, y, z) = (2x, 2y, -e^z)$
Evaluating this gradient at $(0, 1, 0)$, we get: $\nabla F(0, 1, 0) = (0, 2, -1)$
So, the normal vector to the tangent plane at $(0, 1, 0)$ is $(0, 2, -1)$. Using the point-normal form of the equation of a plane, the equation of the tangent plane is: $2(y - 1) - z = 0$
d) Parameterization of the Surface Obtained by Rotating $C$:
Consider the curve $C = \{(x, y, z) \in \mathbb{R}^3 \,|\, z = y^4\}$. I want to parameterize the surface $S$ that is obtained by rotating $C$ around the z-axis.
I suggest the following parameterization: $S(r, \theta, z) = (r \cos(\theta), r \sin(\theta), z^{1/4})$
Here, $r$ and $\theta$ define the position in the xy-plane, and $z$ is determined by $z = y^4$, so $y = z^{1/4}$.
Please provide help, feedback and improvements if needed. Thank you!
Your answers for the first three parts are fine.
In part (d), the given set does not describe a curve in $\Bbb R^3$ but a surface. To picture it yourself, imagine the $y,z$ plane and plot the curve $z=y^4$. This equations holds for all $x$, so if you add the $x$ axis to your picture you "stretch" the curve out to get a sort of parabolic cylinder:
If we fix $x=0$, the surface "shrinks" back down to a curve, which is probably what the exercise intended considering the axis of revolution:
Take this to be $C$, and consider the half with positive $y$ coordinate. Rotating any point on $C$ about the $z$ axis traces out a circle of radius equal to $y$ in the plane $z=y^4$. (One such circle is shown at $z=0.5$.)
Now introducing $\theta$, we can parameterize by
$$S(y,\theta) = \left(y \cos \theta, y \sin \theta, y^4\right)$$