AlphaMale28
Apr14-09, 12:14 PM
Dear Sirs and Madam's
I have following problem which I hope you go assist me in. I have been recommended this forum because I heard its the best place with the best science experts in the world.
Anyway the problem is as follows
1. The problem statement, all variables and given/known data
Let A \subset S be a subset of regular surface S. Prove that A itself is a regular surface iff A is open in S. Where A = U \cap S and where U is open in \mathbb{R}^3
I am using a pretty old book by a guy named Do Carmo so just as now. on page 52 there is a definition of a regular surface:
A subset of S \subset \mathbb{R}^3 is a regular surface if for eac p \in S there exist a neighbourhood V in \mathbb{R}^3 and a map x: U \rightarrow V \cap S of a open set U \subset \mathbb{R}^2 onto V \cap S \subset \mathbb{R}^3
Such that
1) x is differentiabel
2) x is an homomorphism.
3) For each q in U the differential dx_q : \mathbb{R}^2 \rightarrow \mathbb{R}^3 is onto-one.
2. Relevant equations
3. The attempt at a solution
condition 2) By the definition above let p \in A. Next assume that x: U \rightarrow S . Where U is open subset of \mathbb{R}^3. Then
x^{-1}(A \cap x(U)) \subset U is a regular surface and by the definition its open in S.
condition 3)
Again we assume that p \in A Next x: u \rightarrow A where U is a subset of \mathbbb{R}^3. next we assume that x(q) = p and that
dxq: \mathbb{R}^2 \rightarrow \mathbb{R}^3 and is thusly one-to-one.
How does this look???
I have following problem which I hope you go assist me in. I have been recommended this forum because I heard its the best place with the best science experts in the world.
Anyway the problem is as follows
1. The problem statement, all variables and given/known data
Let A \subset S be a subset of regular surface S. Prove that A itself is a regular surface iff A is open in S. Where A = U \cap S and where U is open in \mathbb{R}^3
I am using a pretty old book by a guy named Do Carmo so just as now. on page 52 there is a definition of a regular surface:
A subset of S \subset \mathbb{R}^3 is a regular surface if for eac p \in S there exist a neighbourhood V in \mathbb{R}^3 and a map x: U \rightarrow V \cap S of a open set U \subset \mathbb{R}^2 onto V \cap S \subset \mathbb{R}^3
Such that
1) x is differentiabel
2) x is an homomorphism.
3) For each q in U the differential dx_q : \mathbb{R}^2 \rightarrow \mathbb{R}^3 is onto-one.
2. Relevant equations
3. The attempt at a solution
condition 2) By the definition above let p \in A. Next assume that x: U \rightarrow S . Where U is open subset of \mathbb{R}^3. Then
x^{-1}(A \cap x(U)) \subset U is a regular surface and by the definition its open in S.
condition 3)
Again we assume that p \in A Next x: u \rightarrow A where U is a subset of \mathbbb{R}^3. next we assume that x(q) = p and that
dxq: \mathbb{R}^2 \rightarrow \mathbb{R}^3 and is thusly one-to-one.
How does this look???