Question

topology:

Prove that the open ball B2: = {(x, y) ∈ R^{2}| x^{2}+ y^{2}<1} in R^{2}is homeomorphic to the open squared unit C2: = {(x, y) ∈R^{2}| 0 <x <1.0 <and <1}

Answer #1

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Write vectors in R2 as (x,y). Define the relation on R2 by
writing (x1,y1) ∼ (x2,y2) iff y1 − sin x1 = y2 − sin x2 . Prove
that ∼ is an equivalence relation.
Find the classes [(0, 0)], [(2, π/2)] and draw them on the
plane. Describe the sets which are the equivalence classes for this
relation.

1. Consider the set U={(x,y) in R2| -1<x<1 and y=0}. Is U
open in R2? Is it open in R1? Is it open as a subspace of the disk
D={(x,y) in R2 | x^2+y^2<1} ?
2. Is there any subset of the plane in which a single point set
is open in the subspace topology?

Prove that the open rectangle in R2
S = { (x,y) | 2 < x<5 -8 < y <
-1}
is an open set in R2, with
the usual Euclidean distance metric.

Prove that there are no rational numbers x and y such that x2
-y2 =1002.

Prove the Basic Principal of Difference of squares: If x2 ≡ y2
(mod n) and x is not ± y, where x and y lie in the range {0, … ,
n-1}, then n is composite and has gcd(x-y, n) as a non-trivial
factor.

Prove the Basic Principal of Difference of squares: If x2 ≡ y2
(mod n) and x is not ± y, where x and y lie in the range {0, … ,
n-1}, then n is composite
and has gcd(x-y, n) as a non-trivial factor.

f(x, y) = x2 + y2 + 2xy + 6.
1. Find all the local extremas.
2. Does the function f has an absolute max or min on R2 ?
3. Draw E = {(x, y) ∈ R2; x >=0; y >=0; x + y<=1}.
4. Explain why f has an absolute max and min on E and find
them.

Solve:
uxx + uyy = 0 in {(x,y) st x2 +
y2 < 1 , x > 0, y > 0}
u = 0 on x=0 and y=0
∂u/∂r = 1 on r=1

Evaluate ∫∫Sf(x,y,z)dS , where f(x,y,z)=0.4sqrt(x2+y2+z2)) and S
is the hemisphere x2+y2+z2=36,z≥0

A lamina occupies the first quadrant of the unit disk
(x2+y2≤1x2+y2≤1, x,y≥1x,y≥1). It's density function is
ρ(x,y)=xρ(x,y)=x. Find the center of mass of the lamina.

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