| course |
course_year |
question_number |
tags |
title |
year |
Geometry |
IB |
14 |
|
1.I $2 \mathrm{~A} \quad$ |
2005 |
Let $\sigma: \mathbf{R}^{2} \rightarrow \mathbf{R}^{3}$ be the map defined by
$$\sigma(u, v)=((a+b \cos u) \cos v,(a+b \cos u) \sin v, b \sin u)$$
where $0<b<a$. Describe briefly the image $T=\sigma\left(\mathbf{R}^{2}\right) \subset \mathbf{R}^{3}$. Let $V$ denote the open subset of $\mathbf{R}^{2}$ given by $0<u<2 \pi, 0<v<2 \pi$; prove that the restriction $\sigma_{V}$ defines a smooth parametrization of a certain open subset (which you should specify) of $T$. Hence, or otherwise, prove that $T$ is a smooth embedded surface in $\mathbf{R}^{3}$.
[You may assume that the image under $\sigma$ of any open set $B \subset \mathbf{R}^{2}$ is open in $T$.]