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1. The problem statement, all variables and given/known data[/b

Let [tex]f\in C^{\infty}[/tex][tex](R\R^{2})[/tex] and let S be the set of points in [tex]R\R^{3}[/tex] given the graph of f. Thus, [tex]s={{(x,y,z=f(x,y))|(x,y)\in[/tex][tex]R\R^{2}}[/tex].

a) Show that this set of points can be viewed as a regular level surface.

b) Let X=(x,y,z) be a point on this surface. Find a basis for the tangent space TxS.

c) Give a cover for this surface.

2. Relevant equations

Ehm. Not really "equations", per se. We will need the coordinate basis for R3 which is [tex]\partial[/tex]x, [tex]\partial[/tex]y, [tex]\partial[/tex]z.

3. The attempt at a solution

I am soo, so confused. I guess our F=f(x,y). To show that something is a regular level surface, I believe that we have to show that not all partial derivatives vanish at the point x on the surface. If we had an actual f(x,y) I would compute F*, the differential, and show that it is not simultaneously 0 at some point x on the surface. But, we don't have an explicit f(x,y). So I have no idea how to show this, or to show equivalently, the the mapping from TxU (for our open set U) to T[tex]_{F(x)}[/tex]S is surjective.

I am TOTALLY lost as to how to come up with a basis. It seems that we should compute the kernel of F* and its basis, but I'm not clear on what we do with this basis.

If I could figure out parts a and b, I MIGHT be able to figure out part c. I know that a cover is the union of surface patches. But then I'd have to start by making coordinate surface patches first... would I be defining some map or something??

I'm terribly confused. :yuck: If anyone can help that would be marvelous!

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# Homework Help: Differential geometry hypersurface problem - need help starting

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