Differential geometry hypersurface problem - starting

[quasar_4]

differential geometry hypersurface problem - need help starting!

1. Homework Statement [/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.

Homework 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.

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. If anyone can help that would be marvelous!

 

[HallsofIvy]

1. Homework Statement [/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.

Homework 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.

The Attempt at a Solution

I am soo, so confused. I guess our F=f(x,y).

No. A "level curve" for a function f(x,y) would be a curve in the xy-plane. In order that a surface in 3 dimensions be a level surface, it must be of the form F(x,y,z)= constant. In this case, given z= f(x,y), you know that z-f(x,y)= 0.

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.

What can you say about the partial derivatives of z-f(x,y)?

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.

Given that the surface is defined by z= f(x,y), any tangent vector is of the form [itex]\vec{i}+ f_x\vec{k}[/itex] or [tex]\vec{i}+ f_y\vec{k}[/itex].

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. If anyone can help that would be marvelous!