Show that the unit cylinder can be covered by a single surface patch, the unit s

Question

Show that the unit cylinder can be covered by a single surface patch, the unit sphere cannot. (The second part requires some point set topic Show that every open subset of a surface is a surface. Show that a curve on the unit cylinder that intersects the straight-line cylinder parallel to the z-axis at a constant angle must be a straight line or a circular helix. Find a surface patch for the ellipsoid x^2/p^2 + y^2/q^2 + z^2/r^2 where p, q and r are non-zero constants. (A picture of an ellipsoid can in Theorem 5.2.2.) 4.1.8Show that theta (u, v) = (sin u, sin v, sin (u + v)), -pi/2

Explanation / Answer

SOLUTION :-

let assume a subset S of R3 is a surface if, for every point pS, there is an open set U in R2 and an open set W in R3 containing p such that SW is homeomorphic to U.

A subset of a surface SS of the form SW, where W is an open subset of R3, is called an open subset of S.

From this we say,

Let W be an open subset of R3. SW is an open subset of S.

For every point pSW, there is an open set U in R2 and an open set W=SW in R3 containing p such that (SW)W=SW is homeomorphic to U by the mapping f :(x,y)(x,y,0).

So, from this we say that SW is a surface.

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