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Currently I am focused on Chapter 3: Differential Manifolds ...

I need help in order to fully understand Example 3.1.13 ...

Example 3.1.13 reads as follows:

My questions are as follows:

**Question 1**In the above Example from Lovett we read the following:

" ... ... The image of \(\displaystyle \vec{X}\) is a half-torus \(\displaystyle M\) with \(\displaystyle y \ge 0\), which to conform to Definition 3.1.11, is easily covered by four coordinate patches ... ... "

Can someone please explain and demonstrate how \(\displaystyle M\) can be covered by four coordinate patches ... ...

**Question 2**Can someone please explain/demonstrate why/how the boundary \(\displaystyle \partial M\) is the disconnected manifold with \(\displaystyle R_+\) and \(\displaystyle R_{ - } \)as the two connected components ... ...

Help will be much appreciated ... ...

Peter

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The above post refers to Lovett Definition 3.1.11 ... so I am providing access to the same ... as follows:

Since Definition 3.1.11 above refers to Definition 3.1.3 I am providing access to the same ... as follows:

Hope that helps ...

Peter