Calculus on Submanifolds Challenge
Two multi-part questions about submanifolds given to my class as challenges based on the book Calculus on Manifolds by Spivak. The more detailed the better.
Answer
Answers can only be viewed under the following conditions:
- The questioner was satisfied with and accepted the answer, or
- The answer was evaluated as being 100% correct by the judge.
1 Attachment
Persimmonl
63
The answer is accepted.
Join Matchmaticians Affiliate Marketing
Program to earn up to a 50% commission on every question that your affiliated users ask or answer.
- answered
- 1200 views
- $50.50
Related Questions
- Is $\int_0^1 \frac{\cos x}{x \sin^2 x}dx$ divergent or convergent?
- Optimisation Problem
- Find the area bounded by the graphs of two functions
- Epsilon-delta definitoon of continuity for $f : x → x^3$
- Determine the surface area of a ball, rotating a function about $x$-axis.
- Prove that $\int_0^1 \left| \frac{f''(x)}{f(x)} \right| dx \geq 4$, under the given conditions on $f(x)$
- Variation of Parameter for Variable Coefficient Equation
- Find the equation of the tangent line through the function f(x)=3x$e^{5x-5} $ at the point on the curve where x=1
What is the formal definition of parameterization you are using in class?
Hey, give me a minute to type it up. Is there anyway I could send a pdf to you? If not there should not be any issue with using wichever definition you want as long as you announce it first.
Well the issue is usually my definition would be a local C^\infty diffeomorphism \phi from an open U in R^n onto an open V inside of M, but this makes the first part of the problem trivial. Maybe you can just type it here? If you know latex I can just read it when it's uncompiled.
If you're using a textbook you could just refer me to the definition so that what I write will be consistent with the notions you are familiar with.
Oh, got it. Here goes: Let E, G be Banach spaces and M ⊂ G a non-empty subset. The following are equivalent. i) M is a C^k E-submanifold of G. ii) For each p ∈ M, there exists a embeddening C^k ϕ : U ⊂ E → G (U is open) such that p ∈ ϕ(U) ⊂ M and ϕ(U) is open of M in the induced topology. ϕ is called a C^k E-local parameterization of M in p.
Great, got it. Can I assume that all manifolds are C^\infty manifolds?
Just checked, we proved that after the challenge was issued. So no, can't assume, unfortunately.
Okay finally: have you proved the inverse function theorem on Banach spaces?
Yes, we did.
Hey, just checking on this. Doing alright? Will you be able to finish it in time?
Any other info you need, just ask.
Yes, it's pretty easy. However I just lost all my progress when the website reloaded so I'll type it up somewhere else and upload it in a couple hours.
As I submitted I noticed a typo: in 1.d.) I write "the embedded close submanifold U on which x_2^2 + y_2^2" but this should read "... on which x_2^2 + y_2^2 = 1". Somehow that got left out.
Another important thing to note: in 2.a.) the conditions "cos(a) \neq 0" and "sin(a) \neq 0" should be switched, that was a typo.
Finally: "parallelization" in 2.c.) is more accurately a local basis of sections at every point, it is not a parallelization.