The definitions of oscillation of a set and point respectively:
\omega_f(A)=\sup\limits_{x,y\in A}f(x)-f(y)|
\omega_f(x)=\inf\{\omega_f(x)=\inf\{\omega_f(x-\epsilon, x+\epsilon)\cap A) : \epsilon > 0\}
Given \epsilon > 0 , suppose \omega_f(x) < \epsilon for each x \in [a,b] . Then show there is \delta > 0 such that for every closed interval I \in [a,b] with l(I)< \delta we have \omega_f(I) < \epsilon .
My first approach to this was trying to think of it as an anaglous to the definition...
The question is as follows: Using an induction argument if n > 1 is a natural number then n-1 is a natural number.
P(n)=n-1 such that n-1 is a natural number
Following the steps:
Base case: n=2, P(2)=1 which is a natural number.
We fix a natural number n and assume that P(n)=n-1 is...
Prove that a sequence of uniformly convergent bounded functions is uniformly bounded.
Attempt at proof:
So first we observe the following: ||fn||\leqMn. Each function is bounded. Also, |fn-f|\leq\epsilon for all n \geq N. First off, we observe that for finitely many fn's, we have them...
Let M be a simple surface, that is, one that is the image of a single proper
patch x: D -> R3. If y: D -> N is any mapping into a surface N, show that
the function F: M -> N such that:
F (x(u, v)) = y(u, v) for all (u, v) in D is a mapping of surfaces.
To demonstrate this one does the...
This is probably falls within a problem of Mathematica as opposed to a question on here but I have a question about the following:
Given some cylinder with the shape operator matrix:
{{0,0},{0,-1/r}}
We get eigenvalues 0 and -1/r and thus eigenvectors {0, -1/r} and {1/r, 0} by my...
Homework Statement
Given the following:
Some surface M: z=f(x,y) where f(0,0)=fx(0,0)=fy(0,0)
and
U =-f1U1-fyU2+U3}/Sqrt[1+fx2+fy2]
and
u1 = U1(0)
u2 = U2(0)
are vectors tangent to M at the origin 0.
We want to prove
S(u1)=fxxu1+fxyu2
My problem here is conceptually wadding through this...
I'm having problems understanding surface parametrization from differential geometry.
We are given two general forms for parametrization:
\alpha(u,v) = (u,v,0)
and x(u,v)=(u,v,f(u,v))
This is one I'm especially stuck on:
y=Cosh(x) about the x-axis
\alpha(u,v)=(u, Cosh[v],0)...
http://tinypic.com/r/xqf446/7
^^This is the hint I mentioned.
http://tinypic.com/r/24101at/7
^^This is the problem its referencing.
I'm having trouble figuring out what it wants me to do for part (a).
When it mentions denoting a f and g as follows:
f = U_1.(a/||a||) = (1,0).(a/||a||) =...
This is from O'Neil's differential geometry. I'm having trouble parsing through the problem/hint.
Given any curve a that does not pass through the origin has an orientation-preserving reparameterization in the polar form:
a(t) = (r(t)*Cos(V(t)),r(r)*Sin(V(t)))
where r(t)= ||a(t)||...
Great, it all makes sense. However, how does this bijection prove that O*Gx:K=G. To me it just proves they are equal, but that can't always be the case.
Sort of, but I'm getting kind of thrown off by his notation.
So the function that is defined is sending f: g*H---->Orb(x) where H=G/Gx.
Is this just saying that f(gh)=gx or f(g)=gx because H is everything that doesn't move x so its not really worth mentioning.
So I know this is the orbit-stabilizer theorem. I saw it in Hungerford's Algebra (but without that name).
So we want to form a bijection between the right cosets of the stabilizers and the orbit. Could I define the bijection as this:
f: gG/Gx--->gx
Where H=G/Gx
f(hx)=gx h in H
^ Is that...