STATEMENT: Let $\gamma(t)=(a(t),b(t)),t\in I$(an open interval), be a smooth injective curve in the $xz$-plane, and suppose $a(t)>0$ and $\dot{\gamma}(t)\neq 0$ for all $t\in I$. Let $M\subseteq \mathbb{R}^3$ be the surface of revolution obtained by revolving the image of $\gamma$ about the $z$-axis.
a)Show that $M$ is an immersed sub manifold of $\mathbb{R}^3$, and is embedded if $\gamma$ is an embedding.
b)Show that the map $\varphi(\theta,t)=(a(t)\cos(\theta),a(t)\sin(\theta),b(t))$ from $\mathbb{R}\times I$ to $\mathbb{R}^3$ is a local parametrization of $M$ in a neighborhood of any point.
c)Compute the expression for the induced metric on $M$ in $(\theta,t)$ coordinates.
d)Specialize this computation to the case of the doughnut-shaped torus of revolution give by $(a(t),b(t))=(2+\cos(t),\sin(t))$
QUESTION So I wanted someone to check if my solution is correct.
a)We note that if $\varphi:S^1\times I\rightarrow\mathbb{R}^3$ then $$d\varphi=\begin{bmatrix}a'(t)\cos\theta& -a(t)\sin\theta\\ a'(t)\sin\theta&a(t)\cos\theta\\ b'(t)& 0 \end{bmatrix}$$
So our map is an immersion since the two columns are always linearly independent. The only thing we should verify is that $M\cong S^1\times I$. This is easy to verify as we can take the map $\phi:S^1\times I\rightarrow M$ where $\phi(x,y,t)=(a(t)x,a(t)y,b(t))$, which is a diffeomorphism if we check it in local coordinates.
b)From considering $\varphi$ as we did in part $a$ we can restrict it to open sets of $S^1$ which products an open image since the map is a diffeomorphism.
c)$$ \begin{align} g=&(d\varphi^1)^2+(d\varphi^2)^2+(d\varphi^1)^2\\ =&(a'(t)\cos\theta dt-a(t)\sin\theta d\theta)^{2}+(a'(t)\sin\theta dt+a(t)\cos\theta d\theta)^{2}+(b'(t)dt)^{2}\\ =&\left(a'(t)^{2}+b'(t)^{2}\right)\left(dt\right)^{2}+a^{2}(t)\left(d\theta\right)^{2} \end{align} $$
d)$$g=(dt)^{2}+(2+\cos t)^{2}(d\theta)^{2}$$