(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Charts on the torus T.

Let S_{1}be the unit circle, and for each value 0 ≤ θ < 2π, let P(θ) be the point on the circle

at angle θ.

Let S_{1}×S_{1}be the Cartesian product of two circles. The elements of S_{1}×S_{1}

are P(φ), P(θ), where 0 ≤ φ, θ < 2π.Let P(φ, θ) be the point on the torus at angles φ and θ, i.e., P(φ, θ) = ((R + r cos(φ)) cos(θ),(R + r cos(φ)) sin(θ), r sin(φ)).

Deﬁne a map δ : T → S_{1}× S_{1}

by δ(P(φ, θ)) = (P(φ), P(θ)).

Let M^{φ}_{N}and M^{φ}_{S}be the charts on the ﬁrst S_{1}

and let M^{θ}_{N}and M^{θ}_{S}be the charts on the second S_{1}.

Putting the ﬁrst charts together with the second charts, we get 4 charts on S_{1}× S_{1}.

1) Use the 4 charts on S_{1}XS_{1}to define 4 charts on T.

2) For each of the 4 charts on T, describe points on T that are not on the chart

2. Relavent information

M_{S}from the circle minus the south pole S to the x-axis that take a point P on the circle to the intersection of the line from the south pole (0, −1) through P with the x-axis, and the second M_{N}from the circle minus the north pole N to the x-axis that take a point P on the circle to the intersection of the line from the north pole (0, 1) through P with the x-axis.

3. The attempt at a solution

Ok, so I still don't understand how charts work. I missed the class when the professor talked about charts, and I've looked them up online, but I don't understand how to define charts and in particular, I don't understand how to define these charts. To be completely honest, I have no clue where to even start with this. My "charts" for M seem to have nothing at all to do with the taurus, and I have no clue how to relate them (let alone relating 2 sets of charts relating M to M and M x M to T). If somebody could maybe explain this to me or at lease give me a push in the right direction, it would be GREATLY appreciated. Also, if you could explain how to show a chart (I'm guessing he doesn't literally mean a chart of all possible values, but I don't know what charts look like)

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# Homework Help: Charts of a Torus

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