Probability Theory, Math 170a, Fall 2012, Toni Antunovi - Homework 1
c
c
solutions
From the textbook solve the problems 2 (parts b and c), 10, 54, 56 at the end of the Chapter 1.
Solution to problem 2:
(b) If x (A B )c . Then x A B and so either x A or x
1.6 Line Integrals and Green's Theorem 73
DEFINITION A function u(z) = u(x, y) with continuous ﬁrst and second partial derivatives
with respect to both x and y is harmonic on an open set D if
dzu dzu
A =— —=
u iix2+6y2
0 on D.
THEOREM 2 Suppose that u is
Homework 1 Math 170
Reader: Michael Pejic mpejic@math.berkeley.edu
Problem set: Franklin Section 1.2, Problems 1,2,3,5,6,9,10,14,15,16
Problems checked: Franklin Section 1.2, Problems 1,2,3,5,6,14,15
Grading scheme:
X
O
for complete: signicant eort demons
Homework 6 Math 170
Reader: Michael Pejic mpejic@math.berkeley.edu
Problem set: Franklin Section 1.11, Problems 1,2,5; Section 1.12, Problems 1,2,5,6
Problems checked: Franklin Section 1.11, Problem 5; Section 1.12, Problems 5,6
Grading scheme:
X
O
for co
Homework 8 Math 170
Reader: Michael Pejic mpejic@math.berkeley.edu
Problem set:
Franklin Section 1.15, Problems 1,2,3,4,7
Problems checked:
Franklin Section 1.15, Problems 1,2
Grading scheme:
X for complete": signicant eort demonstrated
O for fail": lack
Homework 10 Math 170
Reader: Michael Pejic mpejic@math.berkeley.edu
Problem set:
Franklin Section 1.17, Problems 1,2,3,4,5,6,9,11
Problems checked:
Franklin Section 1.17, Problems 1,2,3,5,6
Grading scheme:
X for complete": signicant eort demonstrated
O fo
Homework 13 Math 170
Reader: Michael Pejic mpejic@math.berkeley.edu
Grading scheme:
3
for excellent:
Necessary steps are all shown and well explained.
Solution is correct.
2
for fair:
Necessary steps are all shown.
There are minor gaps in explanantion and
Homework 11 Math 170
Reader: Michael Pejic mpejic@math.berkeley.edu
Problem set: Franklin Section 1.18, Problems 1,2,3,4
Problems checked: Franklin Section 1.18, Problems 2,4
Grading scheme:
X
O
for complete: signicant eort demonstrated
for fail: lack of
Homework 12 Math 170
Reader: Michael Pejic mpejic@math.berkeley.edu
Problem set:
Franklin Section 2.1, Problems 1,2,3,4,5,13,14,15,16,17,18
Problems checked:
Franklin Section 2.1, Problems 1,2,4,13,14,15,18
Grading scheme:
X for complete": signicant eort
Homework 5 Math 170
Reader: Michael Pejic mpejic@math.berkeley.edu
Problem set: Franklin Section 1.10, Problems 1,2,3,4,5,6
Problems checked: Franklin Section 1.10, Problems 1,5,6
Grading scheme:
X
O
for complete: signicant eort demonstrated
for fail: lac
1.5 The Exponential, Logarithm, and Trigonometric Functions 53
provided an appropriate branch of the logarithm is chosen. A careful examination
of the mapping w = sin 2 (see the exercises) shows that sin 2 maps the strip
_ 7t 1t
{x+1y: —§<x<~2-,—oo<y<oo}
1.6 Line Integrals and Green's Theorem 71
oriented and y1 is oriented clockwise. Then, just as above,
6x ﬂy
so Green’s Theorem implies that
J dz +J dz :0-
Yz_p Viz—p
Reversing the orientation of yl so that it is now traversed counterclockwise, we
conclude
1.6 Line Integrals and Green's Theorem 69
This establishes the formula
for any triangle 1‘ and its inside Q. A similar argument, using triangles with a vertical
edge, establishes the formula
I @dxdy= —J udx.
nay 1‘
Thus, Green’s Theorem is veriﬁed for a t
1.6 Line Integrals and Green's Theorem 65
on the curve but only on p, q, and m. Indeed, a glance at the computations in
Example 11 shows that
b 1 1 t=b
m _ m I = m+
L2 dz — v (t)v(t)dt m + 1v (0 1:“
1
= m + 1 [v"‘“(b) - v’"“(a)]
= 1 [qm+1 _ pm+1]
m
1.6 Line Integrals and Green's Theorem 67
R
P=a+ib
Q=c+ib
R=d+ie
P Q
Figure 1.32
Example 12* Verify Green’s Theorem for a triangle.
Solution Suppose first that the triangle 1‘ has one horizontal side (see Fig. 1.32).
Let 0 be a function that has continuou
1.6 Line Integrals and Green's Theorem 63
Example 8 Estimate
where y is the semicircle Re‘“, —1z s 6 S 0, R > 2.
Solution On the semicircle,
1
zz+4
1
Rz—m
S
since lz2 + 4| 2 |z|2 — 4 by the triangle inequality (see Section 2). The length of
1.6 Line Integrals and Green's Theorem 57
disjoint open connected sets, one bounded and the other unbounded. The bounded
piece is the inside of the curve and the unbounded piece the outside. Despite the
almost painful obviousness of this statement, the th
1.6 Line Integrals and Green's Theorem 61
inequality
r g(t) (112
The inequality is obviously true if g(t) dt = 0, so we may assume that If: 90:) dt 9'5 0.
Let
b
6 = Arg<J g(t) (it)
and deﬁne h(t) = e‘wga), a s t s b. Then
I!) g(t) dt
= r e“°g(t) dt = J1
1.5 The Exponential, Logarithm, and Trigonometric Functions 55
Show that the following identities hold:
(i) cosh2 (z) — sinh2 (z) = 1
(ii) cosh z = cos(iz)
(iii) sinh z = —i sin(iz)
(iv) |cosh 212 = sinh2 x + cos2 y
(v) lsinh 212 = sinh2 x + sin2 y
22. Sh
1.6 Line Integrals and Green's Theorem 59
Figure 1.30
where t increases from a to R. In summary, then,
z=Re“’, 5<6<2n—6
z=te“2"“”, R2t25
y. z=£ei°, 211—62626
z=tei", sStSR. El
Each curve )1 is oriented by increasing t. The curve 3) begins at y(a), is tr
1.5 The Exponential, Logarithm, and Trigonometric Functions 51
ix— _ix-
e 1e Y1_e zen,
so y1 = y2 and x1 — x2 is an integer multiple of 21:. This again implies that x1 = x2.
Let us now ﬁnd the range of f(z) = sin 2 on the strip 0 s x g n/2 and
0 s y < 00.
Homework 9 Math 170
Reader: Michael Pejic mpejic@math.berkeley.edu
Problem set:
Franklin Section 1.16, Problems 1,2,3,7,10
Problems checked:
Franklin Section 1.16, Problems 1,2
Grading scheme:
X for complete": signicant eort demonstrated
O for fail": lack
Homework 3 Math 170
Reader: Michael Pejic mpejic@math.berkeley.edu
Problem set: Franklin Section 1.5, Problems 1,2,4,5,10; Section 1.4, Problems 1,2,3,5
Problems checked: Franklin Section 1.3, Problems 1,2; Section 1.4, Problems 1,2,3
Grading scheme:
X
O
Math 170 - Mathematical Methods for Optimization - Fall 2014
Announcements and Handouts
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Wednesday 10 December, 1-2:30 in 959 Evans. Monday 15 December, 11:0012:30 in 891 Evans. Tuesda