Question

Let d be the usual metric on R2. (a) Find d(x,y) when x = (3,−2) and y = (−3,1). (b) Let x = (5,−1) and y = (−3,−5). Find a point z in R2 such that d(x,y) = d(y,z) = d(x,z). (c) Sketch Br(a) when a = (0,0) and r = 1. The Euclidean metric is by no means the only metric on Rn which is useful. Two more examples follow (9.2.10 and 9.2.12).

Answer #1

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.

Let (X,d) be a metric space which contains an infinite countable
set Ewith the property x,y ∈ E ⇒ d(x,y) = 1.
(a) Show E is a closed and bounded subset of X. (b) Show E is
not compact.
(c) Explain why E cannot be a subset of Rn for any n.

Let (X, d) be a compact metric space and F: X--> X be a
function such that d(F(x), F(y)) < d(x, y). Let G: X --> R be
a function such that G(x) = d(F(x), x). Prove G is continuous
(assume that it is proved that F is continuous).

Let y = x 2 + 3 be a curve in the plane.
(a) Give a vector-valued function ~r(t) for the curve y = x 2 +
3.
(b) Find the curvature (κ) of ~r(t) at the point (0, 3). [Hint:
do not try to find the entire function for κ and then plug in t =
0. Instead, find |~v(0)| and dT~ dt (0) so that κ(0) = 1 |~v(0)|
dT~ dt (0) .]
(c) Find the center and...

1. Let f : R2 → R2, f(x,y) = ?g(x,y),h(x,y)? where g,h : R2 → R.
Show that
f is continuous at p0 ⇐⇒ both g,h are continuous at p0

Suppose K is a nonempty compact subset of a metric space X and
x∈X.
Show, there is a nearest point p∈K to x; that is, there
is a point p∈K such that, for all other q∈K,
d(p,x)≤d(q,x).
[Suggestion: As a start, let S={d(x,y):y∈K} and show there is a
sequence (qn) from K such that the numerical sequence (d(x,qn))
converges to inf(S).] Let X=R^2 and T={(x,y):x^2+y^2=1}.
Show, there is a point z∈X and distinct points a,b∈T
that are nearest points to...

Let D be the solid region defined by D = {(x, y, z) ∈ R3; y^2 +
z^2 + x^2 <= 1},
and V be the vector field in R3 defined by: V(x, y, z) = (y^2z +
2z^2y)i + (x^3 − 5^z)j + (z^3 + z) k.
1. Find I = (Triple integral) (3z^2 + 1)dxdydz.
2. Calculate double integral V · ndS, where n is pointing
outward the border surface of V .

Each of the following deﬁnes a metric space X which is a subset
of R^2 with the Euclidean metric, together with a subset E ⊂ X. For
each,
1. Find all interior points of E,
2. Find all limit points of E,
3. Is E is open relative to X?,
4. E is closed relative to X?
I don't worry about proofs just answers is fine!
a) X = R^2, E = {(x,y) ∈R^2 : x^2 + y^2 = 1,...

Let (X, d) be a metric space, and let U denote the set of all
uniformly continuous functions from X into R. (a) If f,g ∈ U and we
define (f + g) : X → R by (f + g)(x) = f(x) + g(x) for all x in X,
show that f+g∈U. In words,U is a vector space over R. (b)If f,g∈U
and we define (fg) : X → R by (fg)(x) = f(x)g(x) for all x in X,...

Let (X,d)
be a complete
metric space, and T
a d-contraction
on X,
i.e., T:
X
→ X
and there exists a q∈
(0,1) such that for all x,y
∈ X,
we have d(T(x),T(y))
≤ q∙d(x,y).
Let a
∈ X,
and define a sequence (xn)n∈Nin
X
by
x1 :=
a
and ∀n ∈
N: xn+1
:= T(xn).
Prove, for all n
∈ N,
that d(xn,xn+1)
≤ qn-1∙d(x1,x2).
(Use
the Principle of Mathematical Induction.)
Prove that (xn)n∈N
is a d-Cauchy
sequence in...

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