Question

(Differential Topology)

Show that the projection map from a tangent space T_a_A to its manifold A is a submersion

Answer #1

We will use the local submersion theorem here.

This is a standard proof.

(a)Show that the digital line can be obtained as a
quotient space that results
from a partition of R in the standard topology.
(b) Show that the digital plane, introduced in Section 1.4, can be
obtained as a
quotient space that results from a partition of R2 in the standard
topology.

If X is a topological space, and A ⊆ X, define the subspace
topology on A inherited from X.

Show that a weakly open subset U of a Banach space is also open
in the norm topology.

Show that the following collections τ of subsets of X form a
topology in the given space.
a) Let X = R with τ consisting of all subsets B of R such that
R\B contains finitely many elements or is all of R.
b) Let X = {a, b, c} and τ = {∅, {c}, {a, c}, {b, c}, {a, b,
c}}.

Let T be the half-open interval topology for R, defined in
Exercise 4.6.
Show that (R,T) is a T4 - space.
Exercise 4.6
The intersection of two half-open intervals of the form [a,b) is
either empty or a half-open interval. Thus the family of all unions
of half-open intervals together with the empty set is closed under
finite intersections, hence forms a topology, which has the
half-open intervals as a base.

Show that projection of a line, from any finite point P, onto a
parallel line is represented by a function of the from f(x) =
ax+b.

Let X be a topological space with topology T = P(X). Prove that
X is finite if and only if X is compact. (Note: You may assume you
proved that if ∣X∣ = n, then ∣P(X)∣ = 2 n in homework 2, problem 2
and simply reference this. Hint: Ô⇒ follows from the fact that if X
is finite, T is also finite (why?). Therefore every open cover is
already finite. For the reverse direction, consider the
contrapositive. Suppose X...

Let (V, C) be a finite-dimensional complex inner product
space.
We recall that a map T : V → V is said to be normal if
T∗ ◦ T = T ◦ T∗ .
1. Show that if T is normal, then |T∗(v)| = |T(v)|
for all vectors v ∈ V.
2. Let T be normal. Show that if v is an eigenvector of T
relative to the eigenvalue λ, then it is also an eigenvector of
T∗ relative to...

Topic: Calculus 3 / Differential Equation
Q1) Let (x0, y0,
z0) be a point on the curve C described by the following
equations
F1(x,y,z)=c1 , F2(x,y,z)=c2 .
Show that the vector [grad F1(x0,
y0, z0)] X [grad F2(x0, y0,
z0)] is tangent to C at (x0, y0,
z0)
Q2) (I've posted this question before but
nobody answered, so please do)
Find a vector tangent to the space circle
x2 + y2 + z2 = 1 , x + y +...

Use the cobweb diagram to show that the map f(x) =e^-x
has a unique fixed point. Also determine the stability character of
the fixed point from the diagram.

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