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

Consider the ellipsoid L [itex]\subset[/itex]E3 specified by

(x/a)^2 + (y/b)^2 + (z/c)^2=1

(a, b, c [itex]\neq[/itex] 0). Define f: L-S[itex]^{2}[/itex] by f(x, y, z) = (x/a, y/b. z/c).

(a) Verify that f is invertible (by finding its inverse).

(b) Use the map f, together with a smooth atlas of S[itex]^{2}[/itex], to construct a smooth atlas of L.

2. Relevant equations

3. The attempt at a solution

For part (a), would the inverse be f[itex]^{-1}[/itex](x/a, y/b, z/c)= (x,y,z)?

So that you take the points on the ellipsoid and get points on S[itex]^{2}[/itex]?

For (b), a smooth atlas of S[itex]^{2}[/itex] is

U[itex]_{1}[/itex]= {(x,y,z)[itex]\in[/itex]S[itex]^{2}[/itex]|(x,y,z)[itex]\neq[/itex](1,0,0)}

U[itex]_{2}[/itex]= {(x,y,z)[itex]\in[/itex]S[itex]^{2}[/itex]|(x,y,z)[itex]\neq[/itex](-1,0,0)}

But how do I use that with f to form an atlas of L?

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# Homework Help: Differential geometry: smooth atlas of an ellipsoid

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