Hw: Page 55, 84, 8.7
Du Fri Feb 3
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4
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Welly #f
A is an open set of R" vach, 3rzo such that
BRICA
Bis a closed set of il Re is open
R and 4 ure both open
clased
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proposition 2.1
A is closed A Contains.
all of its linit points
let Ā=AU { limit points of A} Then
A is closed as A
Proof Idea: A is closed, R"-A isopen.
R". A Cannot have
any
limit points oft.
Thcare 8.2:
FER f is continuous illes v open set
un R, Plül is open in R"
fis Continuous y closeal set cine,
.. (C) is
closed in R".
prove:
f is cortinicous. We info)
since U is open, 3 ro such that B l frat) Cu.
fis Continuous 3870 such that
1-21 B(2) CPU)
F"(U) is open
C is closed in R R c is open f (Ruch is open.
f(R") f(a) a Raf (c) is open focus is closed.
Extern value theorem
a Contacons Function on [a, b] always attains the
meximum and
Values
من اما
minimum
Compact sets:
M
A set A of P" is compact iff eveny infinite
subset of A has a limit point of A
b.
Vous
Note: A is finite A is compact.
Bounded sets
A set 4 of Rh is bounded if 3 B (0) A
Lem 23
V
Lemm 8.4
©
compact set is closed and bounced
ek
[a, b] is Compact in R. (noora abs to)
a Da{{xiysexty"{1} is compaction ?
A closed subset of a compact set is compact
Lewn 8.5
A is compact in R", Bis Compact inte
AXB is Compact in phtm
Ex [115] is Compact inR
[216] is compact inR
[115] x [206] is Compact in p²
Thm 2.6
A is compact A is closed and bounded
A is compact, F is Continuous, then F(A) is compact
Thm 8.8
f ŏ R is Continuous
o is compact. Then there exist a 5 GD
such that fla) = f(2) f(b) for all tino
5
RP
فيما :H
Page 61-63
(6)
1.1, 1.9,1.12
Reted Pili ya klast aftur dices adya Pixada
FR²R za frys , d flerys dzef (eng) dx + fylmaysaly
yofa) f'lay: him fath) flas
him flash) - flag-flash
ht
h
of (h): f(ath) - f(a).
a falha f'cash a differential off at paint a
bei einem Schiffocus
sfachs can be approximated by difor Ch) when his small
dfa RR 25 (%Fla) z is near Pandian
FIRR (curves in R)
B o
far lim feath) - film)
FC
LF-Carh).
mestris
fia) - EUR)
flas is the
Velocity vector in
fon (w
I flail: speed
df (x) = f(a)x is linear
fois ami
EX FER: Flt): [ sint] find velocity and speed
t.
F(
.ts
iese []
Cost
f ( 1 ] .
F43300
F91-1)=(9)
4: RR
6,9:
RR
* (F+ g3 a fitgh
ELF Where is
* (0f) =øft of
* 0, there
exists N such that n 21 a. -al < c.) The equivalence of this statement
and the definition is just a matter of language (Exercise 8.7).
Examples: (a) is not compact, because the set of all integers is an
infinite subset of that has no limit point at all. Similarly, st" is not compact.
(b) The open interval (0, 1) is not compact, because the sequence (1/n); is
an infinite subset of (0, 1) whose limit point is not in the interval. Similarly,
open balls fail to be compact. (c) If the set F is finite, then it is automatically
compact because it has no infinite subsets which could cause problems.
Closed intervals do not appear to share the problems in regard to compact-
ness) of open intervals. Indeed the Bolzano-Weierstrass theorem says precisely
that every closed interval is compact (see the Appendix). We will see presently
that every closed ball is compact. Note that a closed ball is both closed and
bounded, meaning that it lies inside some ball B,(0) centered at the origin.
the points {xa}i (why?).
Since C is compact, we may assume (taking a subsequence if necessary) that
the sequence {xa}iº converges to a point x, € C. But then X, EVA for some k,
and since the set V is open, it must contain infinitely many elements of the
sequence {xa}i. This contradiction proves the theorem.
Exercises
8.1 Verify that the collection of all open subsets of satisfies conditions (1)-(iii).
8.2 Verify that the collection of all closed subsets of satisfies conditions (19-(iii). Hint: If
(42) is a collection of subsets of Sp", thens-UA-n (p" - ) and
- 4.-U(ST" - A.).
8.3 Show, directly from the definitions of open and closed sets, that open and closed balls
are respectively open and closed sets.
8.4 Complete the proof of Theorem 8.2.
8.5 The point a is called a boundary point of the set A if and only if every open ball centered at
a intersects both A and 3* - A. The boundary of the set A is the set of all of its boundary
points. Show that the boundary of A is a closed set. Noting that the sphere S.(p) is the
boundary of the ball B.(P), this gives another proof that spheres are closed sets.
8.6 Show that is the only nonempty subset of itself that is both open and closed. Hint: Use
the fact that this is true in the case -1 (see the Appendix), and the fact that is a
union of straight lines through the origin.
8.7 Show that is compact if and only if every sequence of points of A has a subsequence
that converges to a point of A.
8.8 If b> for each , show that the sequence {b.) has no limit.
8.9 Prove that the union or intersection of a finite number of compact sets is compact.
8.10 Let (1.)r be a decreasing sequence of compact sets (that is, 41 A, for all n). Prove
that the intersection no. 4, is compact and nonempty. Give an example of a decreasing
sequence of closed sets whose intersection is empty.
8.11 Given two sets Cand Din Rp", define the distance d(C, D) between them to be the greatest
lower bound of the numbers a-b for a € C and be D. If a is a point of Stand D is a
closed set, show that there exists d e D such that da, D)= a --d. Hint: Let B be an
appropriate closed ball centered at a, and consider the continuous function f: BD
defined by f(x) - x-al.
8.12 If C is compact and D is closed, prove that there exist points c e C and d e D such that
d(C, D) - c-dl. Hint: Consider the continuous function : C-3" defined by
f(x) = (x, D).
2 Directional Derivatives and the Differential
63
(b) Show that
(6).
gla) df. -f(a) dg.
(gla))
if gla) 0.
1.8 Let y(t) be the position vector of a particle moving with constant acceleration vector
y"(t)= a. Then show that y(t) - fra + Io + po, where po = y(0) and v. -70). If
1-0, conclude that the particle moves along a straight line through po with velocity
vector Vo (the law of inertia).
1.9 Let y: ** be a differentiable curve. Show that y(t) is constant if and only if y(t)
and y(t) are orthogonal for all t.
1.10 Suppose that a particle moves around a circle in the plane , of radius r centered at 0,
with constant speed u. Deduce from the previous exercise that y(t) and y' (t) are both
orthogonal to y' (t), so it follows that y'(1) -k(ty(t). Substitute this result into the equation
obtained by differentiating y(t):7(0) - 0 to obtain k = -b/r2. Thus the acceleration
vector always points towards the origin and has constant length v/r.
1.11 Given a particle in pp with mass m and position vector y(i), its angular momentum vector
is L(t) y(t) x my' (t), and its torque is T(1)-(1) my"().
(a) Show that L'(1) - T(!), so the angular momentum is constant if the torque is zero
(this is the law of the conservation of angular momentum).
(b) If the particle is moving in a central force field, that is, y(t) and y(t) are always
collinear, conclude from (a) that it remains in some fixed plane through the origin.
1.12. Consider a particle which moves on a circular helix in with position vector
(1)-(a cos wi, a sin wr, but).
(a) Show that the speed of the particle is constant.
(b) Show that its velocity vector makes a constant nonzero angle with the s-axis.
(c) If t, -0 and 12 - 2/w, notice that y(t)-(a, 0, 0) and (t2) -(a, 0,2mb), so the
vector (ta) -(t.) is vertical. Conclude that the equation
(12)-(1)-(2-1 (T)
cannot hold for any TE1, 12). Thus the mean value theorem does not hold for vector-
valued functions.
2 DIRECTIONAL DERIVATIVES AND THE DIFFERENTIAL
We have seen that the definition of the derivative of a function of a single
variable is motivated by the problem of defining tangent lines to curves. In a
mmilar way the concept of differentiability
for functions of several variables is
motivated by the problem of defining tangent planes to surfaces.
function f :
as a
Exercises
1.1 Let S.+** be a differentiable mapping with f'() #0 for all t€ R. Let p be a fixed
point not on the image curve of fas in Fig. 2.4. If q = f(t) is the point of the curve closest
to p, that is, if |p-415|p-fo for all tes, show that the vector p -q is orthogonal
to the curve atq. Hint: Differentiate the function (t) = 1p-fo?
1'110
Figure 2.4
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