# differential geometry: smooth atlas of an ellipsoid

by gotmilk04
Tags: atlas, diff geometry, geometry, manifold, smooth
 P: 45 1. The problem statement, all variables and given/known data Consider the ellipsoid L $\subset$E3 specified by (x/a)^2 + (y/b)^2 + (z/c)^2=1 (a, b, c $\neq$ 0). Define f: L-S$^{2}$ 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$^{2}$, to construct a smooth atlas of L. 2. Relevant equations 3. The attempt at a solution For part (a), would the inverse be f$^{-1}$(x/a, y/b, z/c)= (x,y,z)? So that you take the points on the ellipsoid and get points on S$^{2}$? For (b), a smooth atlas of S$^{2}$ is U$_{1}$= {(x,y,z)$\in$S$^{2}$|(x,y,z)$\neq$(1,0,0)} U$_{2}$= {(x,y,z)$\in$S$^{2}$|(x,y,z)$\neq$(-1,0,0)} But how do I use that with f to form an atlas of L?

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